p Week II Kinematics Viii

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PHYSICS MASTER IIT ACADEMY, Vijayawada Study Material

KINEMATICS

Before dealing with the subject let us have a brief understanding of the terms used in this lesson.

1. MATTER :- Everything in space which can be seen, touched or can be perceived is called as matter.2. SPACE :- Something limited and measurable in length, width or depth and regarded as not filled up.3. MATERIAL :- Anything from which something is or may be made.4. OBJECT :- A thing ( that need not or cannot be named )5. BODY :- A piece of matter that has no regular dimensions.

6. PARTICLE :- A piece of matter smaller than, and part of, an atom.7. POSITION :- The place where something is or stands in relation to the other objects or places.8. LOCATION :- A place or position ( a particular ) in space.9. LENGTH :- The measure from one end to the other end or of the longest side of something.10. PATH :- The line joining the successive positions of a moving body.11. REST :- A body is said to be at rest if it does not change its position with time with respect to the

surroundings.12. MOTION :- A body is said to be in motion if it changes its position with time with respect to the

surroundings.

To study the motion of a body means to determine how its position changes with time. If this is known wecan determine the position of the body at any instant of time. This is the essence of the basic problem ofmechanics. Mechanics ,the study of motion of objects is the oldest of the physical sciences is defined as the study

of the way matter and forces interact with each other. Here we are concerned with rigid macroscopic bodies, i.e.,bodies that you can easily see, do not bend, and are in the solid state.

Statics is a field within mechanics which concerns itself with forces when no change in momentum occurs.

Dynamics is a field concerned with forces and matter when a change in momentum does occur.

Kinematics is a study of motion without regard to the forces present. It is simply a mathematical way todescribe motion. Kinematics is the study of how things move. Here, we are interested in the motion of normalobjects in our world. A normal object is visible, has edges, and has a location that can be expressed with (x, y, z)coordinates.

One dimensional motion is motion along a straight line.

Bodies can be in quite diverse types of mechanical motion : they can move along different trajectories, fasteror slower etc.,. In most actual motions, different points in a body move along different paths. The complete motionis known if we know how each point in the body moves.

We shall begin by discussing the motion of a single particle along a straight line. This is rectilinearmotion. Here motion is considered only along a single axis.

In nature, we often deal with motions whose trajectories are curves and not straight lines, such motionsare called curvilinear. Eg: Planets, artificial satellites, parts of engines and tools, water in rivers, atmospheric air.For a curvilinear motion, we cannot say that only one co-ordinate changes.

Each body has a definite size. Consequently, its different parts have different positions in space. How,then, can we determine the position of a body? In the general case, it is not an easy task. However, in manycases it is not necessary to indicate the position of each point of a moving body. All the points of a boat floating ina river move identically, and so do the points of a suitcase as we lift it from the floor. The position of eachindividual point need not be specified if all the points move identically. The motion of such body is calledtranslational motion. Any imaginary straight line drawn in a body performing translational motion remainsparallel to itself.

A body is said to be in rotatory motion if every particle moves in a curved path about a fixed point.Any motion that repeats itself along the same path in equal intervals of time is calledperiodic motion.If a particle in a periodic motion moves back and forth over the same path, its motion is said to be

oscillatory orvibratory motion.

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PHYSICS MASTER IIT ACADEMY, Vijayawada Study MaterialMotion is relative

According to our modern world-view, it really isn't that reasonable to expect that a special force should berequired to make the air in the train have a certain velocity relative to our planet. After all, the "moving" air in the"moving" train might just happen to have zero velocity relative to some other planet we don't even know about.Aristotle claimed that things "naturally" wanted to be at rest, lying on the surface of the earth. But experiment afterexperiment has shown that there is really nothing so special about being at rest relative to the earth. For instance,if a mattress falls out of the back of a truck on the freeway, the reason it rapidly comes to rest with respect to theplanet is simply because of friction forces exerted by the asphalt, which happens to be attached to the planet.

Rest : If the position of an object does not change as time passes, it is said to be at rest relative toobserver.

Motion : If the position of an object change as time passes, it is said to be in motion relative to observer.(or) continuous change in the position of a body relative to other bodies is called mechanical motion.

Galileo's insights are summarized as follows:

The principle of inertia

No force is required to maintain motion with constant velocity in a straight line, and absolute motiondoes not cause any observable physical effects.

There are many examples of situations that seem to disprove the principle of inertia, but these all resultfrom forgetting that friction is a force. For instance, it seems that a force is needed to keep a sailboat in motion. Ifthe wind stops, the sailboat stops too. But the wind's force is not the only force on the boat; there is also africtional force from the water. If the sailboat is cruising and the wind suddenly disappears, the backward frictionalforce still exists, and since it is no longer being counteracted by the wind's forward force, the boat stops. Todisprove the principle of inertia, we would have to find an example where a moving object slowed down eventhough no forces whatsoever were acting on it.

Self-Check What is incorrect about the following supposed counterexamples to the principle of inertia?

(1) When astronauts blast off in a rocket, their huge velocity does cause a physical effect on their

bodies - they get pressed back into their seats, the flesh on their faces gets distorted, and theyhave a hard time lifting their arms.

(2) When you're driving in a convertible with the top down, the wind in your face is an observablephysical effect of your absolute motion.

Answer (1) The effect only occurs during blastoff, when their velocity is changing. Once the rocketengines stop firing, their velocity stops changing, and they no longer feel any effect. (2) It is onlyan observable effect of your motion relative to the air.

Discussion Questions

A

A passenger on a cruise ship finds, while the ship is docked, that he can leap off of the upper deck and justbarely make it into the pool on the lower deck. If the ship leaves dock and is cruising rapidly, will this passenger

still be able to make it?

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B

You are a passenger in the open basket hanging under a helium balloon. The balloon is being carried along bythe wind at a constant velocity. If you are holding a flag in your hand, will the flag wave? If so, which way?

C Aristotle stated that all objects naturally wanted to come to rest, with the unspoken implication that "rest" wouldbe interpreted relative to the surface of the earth. Suppose we go back in time and transport Aristotle to themoon. Aristotle knew, as we do, that the moon circles the earth; he said it didn't fall down because, likeeverything else in the heavens, it was made out of some special substance whose "natural" behavior was to goin circles around the earth. We land, put him in a space suit, and kick him out the door. What would he expecthis fate to be in this situation? If intelligent creatures inhabited the moon, and one of them independently cameup with the equivalent of Aristotelian physics, what would they think about objects coming to rest?

D The bottle is sitting on a level table in a train's dining car, but the surface of the water is tilted. What can youinfer about the motion of the train?

The line used for this motion is often the familiar x-axis, or x number line. The object may move forward orbackward along this line.

Forward is usually considered positive movement and this movement is usually considered to the right.So, as an object moves forward down the x-axis, it is heading toward larger and larger x coordinates, and we saythat it has a positive displacement and a positive velocity.

Backward is usually considered negative movement to the left. As an object moves backward along the x-axis, it is heading toward smaller and smaller x coordinates, and we say that it has a negative displacement and anegative velocity.

Types of motion

All bodies in motion do not move in the same way. For example : a car moves straight along the road; aspinning top spins round and round; the strings of a Sitar moves up and down and the pendulum clock movessideways in either direction. Thus the motion of the bodies can be classified as

TYPES OF MOTION Motion of objects can be divided into three categories.

(i) TRANSLATIONAL MOTION(ii) ROTATIONAL MOTION(iii) VIBRATIONAL MOTION

TRANSLATIONAL MOTION

"Motion of a body in which every particle of the body is being displaced by the same amount is calledTranslational Motion".

EXAMPLE:(i) Motion of a person on a road.(ii) Motion of a car or truck on a road.

ROTATIONAL MOTION "Type of motion in which a body rotates around a fixed point or axis is called Rotational Motion."

EXAMPLE:

(i) Motion of wheel(ii) Motion of the blades of a fanVIBRATIONAL MOTION

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http://www.vias.org/genchem/persys_he.htmlhttp://www.vias.org/genchem/persys_he.html


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PHYSICS MASTER IIT ACADEMY, Vijayawada Study Material"Type of motion in which a body or particle moves to and fro about a fixed point or mean position iscalled Vibratory Motion."

EXAMPLE:(i) Motion of simple pendulum(ii) Motion of the wires of guitar(iii) Motion of swing

Describing Distance and Time

Center-of-mass motion in one dimension is particularly easy to deal with because all the information aboutit can be encapsulated in two variables: x, the position of the center of mass relative to the origin, and t, whichmeasures a point in time. For instance, if someone supplied you with a sufficiently detailed table of x and t values,you would know pretty much all there was to know about the motion of the object's center of mass.

A point in time as opposed to duration

In ordinary speech, we use the word "time" in two different senses, which are to be distinguished inphysics. It can be used, as in "a short time" or "our time here on earth," to mean a length or duration of time, or itcan be used to indicate a clock reading, as in "I didn't know what time it was," or "now's the time." In symbols, t isordinarily used to mean a point in time, while t signifies an interval or duration in time. The capital Greek letterdelta, , means "the change in...," i.e. a duration in time is the change or difference between one clock reading and another. The notation t does not signify the product of two numbers, and t, but rather one single number, t. If a matinee begins at a point in time t = 1 o'clock and ends at t = 3 o'clock, the duration of the movie was thechange in t,

t = 3 hours - 1 hour = 2 hours .

To avoid the use of negative numbers for t, we write the clock reading "after" to the left of the minussign, and the clock reading "before" to the right of the minus sign. A more specific definition of the delta notation istherefore that delta stands for "after minus before."

Even though our definition of the delta notation guarantees that t is positive, there is no reason why tcan't be negative. If t could not be negative, what would have happened one second before t = 0? That doesn'tmean that time "goes backward" in the sense that adults can shrink into infants and retreat into the womb. It just

means that we have to pick a reference point and call it t = 0, and then times before that are represented bynegative values of t.

Although a point in time can be thought of as a clock reading, it is usually a good idea to avoid doingcomputations with expressions such as "2:35" that are combinations of hours and minutes. Times can instead beexpressed entirely in terms of a single unit, such as hours. Fractions of an hour can be represented by decimalsrather than minutes, and similarly if a problem is being worked in terms of minutes, decimals can be used insteadof seconds.

Self-Check Of the following phrases, which refer to points in time, which refer to time intervals, and which refer totime in the abstract rather than as a measurable number?

(1) "The time has come." (2) "Time waits for no man." (3) "The whole time, he had spit on his chin."Answer (1) a point in time; (2) time in the abstract sense; (3) a time interval

Position as opposed to change in position

As with time, a distinction should be made between a point in space, symbolized as a coordinate x, and achange in position, symbolized as x.

As with t, x can be negative. If a train is moving down the tracks, not only do you have the freedom tochoose any point along the tracks and call it x = 0, but it's also up to you to decide which side of the x = 0 point ispositive x and which side is negative x.

Since we've defined the delta notation to mean "after minus before," it is possible that x will be negative,unlike t which is guaranteed to be positive.

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PHYSICS MASTER IIT ACADEMY, Vijayawada Study MaterialDISTANCE AND DISPLACEMENT

Distance and Displacement refer to traveling from spot to spot. However, distance considers every stepthat you take to be positive. That is, it sums all of the steps you take. Furthermore, it gives you no idea of whatdirection you traveled. It is best used for determining how far you actually traveled on some trip. Displacementdiffers significantly with distance in that it must consider direction. This means that it can have positive andnegative values. It is best used in most circumstances, since it is usually important to know where something isheaded. For translational motion, the variable indicating displacement in the x direction is x, and in the y-direction is y. Thats convenient. The units used in evaluating x or y displacement will most often be meters,although some conversions may be necessary.

A change in position of a particle is called a displacement. ( dis = change; placement = in place ).If a particle moves from position X to position Y, we can represent its displacement by drawing a line from

X to Y; the direction of displacement can be shown by putting an arrowhead ( ) at Y indicating that thedisplacement was from X to Y. The path of the particle need not necessarily be a straight line from X to Y.; thearrow represents only the net effect of the motion, not the actual motion. If we plot a path followed by the particlefrom X to Y, it need not be the same as the displacement XY.

The actual path traveled by a body either in rectilinear or curved path is called the distance travelled bythe body. The length of the path is called the distance traveled. The direction of motion changes from time totime.

A displacement is therefore characterized by a length and a direction. Hence such physical quantities

that behave like displacement are called vectors . Vectors then are quantities that have both magnitude anddirection and combine according to the rules of addition. To sum up : displacement is the shortest distance

between the initial and final position traversed by a body. The length AB is called distance traveled whereas AB .

Some important observations :-

i) Displacement depends on initial and final positions of a body.ii) Displacement does not depend on the path traced by it.iii) Displacement the distance covered by a body.iv) If a body occupies the stating position after covering some distance the displacement is zero i.e., A

body may cover a distance without having a displacementv) The magnitude of displacement is equal to that of distance when the body moves in straight line.vi) A body may not possess displacement without distance covered.

The ratio of distance to displacement in case of a circular motion is 2r : 0 The ratio of distance to displacement in case of a semi - circular motion is : 2. The ratio of distance to displacement in case of a quarter circular motion is 11 : 72.

NUMERICALS

1. A body moves 5m towards E and then 12m towards N. The displacement of the body and the distancetraveled by the body are _________________________________________.

2. An ant moves through a distance of 5m along the length, 4m along the breadth and 3m along the height of arectangular room. Find its displacement.

3. A person moves 30m N, then 20m E and finally 302m S-W. Find his displacement from the original position.

For the ensuing discussion of the equations that govern kinematics we will talk in terms of the x directiononly. The same equations apply to the y direction and we show that by analogy later.

Since kinematics studies the characteristics of moving objects, it begs the question, what will we beconsidering? There are only four quantities to evaluate motion: displacement (distance), velocity (speed),acceleration, and time. The terms in parentheses are the scalar (directionless) equivalents of the preceding terms.

Displacement in the x direction can be determined most simply by looking at the position wheresomething was last seen and comparing it to the position where it was first seen. We call these:

xo = position first seen (sometimes called the starting position), andxf = position last seen (sometimes called the ending or final position).

Displacement is computed by subtracting one from the other.

x = xf - xo.

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Sita is 2 meters to the right of Gita and starts walking. When she was last seen, she was 17 meters to theleft of Gita. What is her last known displacement relative to where she started?

Step 1: Pick a direction as positive. Since Im left-handed and tired of being suppressed by the right-handedmajority, I will choose left as the positive direction,Step 2: Setup equation, plug in numbers, and do the arithmetic. x = xf - xo. = 17 m - (-2 m) = + 19 m

Note that in addition to x and x, there is a third quantity we could define, which would be like anodometer reading, or actual distance traveled. If you drive 10 km, make a U-turn, and drive back 10 km, then yourx is zero, but your car's odometer reading has increased by 20 km. However important the odometer reading isto car owners and used car dealers, it is not very important in physics, and there is not even a standard name ornotation for it. The change in position, x, is more useful because it is so much easier to calculate: to compute x,we only need to know the beginning and ending positions of the object, not all the information about how it gotfrom one position to the other.

Self-Check A ball hits the floor, bounces to a height of one meter, falls, and hits the floor again. Is the xbetween the two impacts equal to zero, one, or two meters?

Answer Zero, because the "after" and "before" values of x are the same.

Frames of reference

The example above shows that there are two arbitrary choices you have to make in order to define aposition variable, x. You have to decide where to put x = 0, and also which direction will be positive. This isreferred to as choosing a coordinate system or choosing a frame of reference. (The two terms are nearlysynonymous, but the first focuses more on the actual x variable, while the second is more of a general way ofreferring to one's point of view.) As long as you are consistent, any frame is equally valid. You just don't want tochange coordinate systems in the middle of a calculation.

Have you ever been sitting in a train in a station when suddenly you notice that the station is movingbackward? Most people would describe the situation by saying that you just failed to notice that the train wasmoving - it only seemed like the station was moving. But this shows that there is yet a third arbitrary choice thatgoes into choosing a coordinate system: valid frames of reference can differ from each other by moving relative to

one another. It might seem strange that anyone would bother with a coordinate system that was moving relative tothe earth, but for instance the frame of reference moving along with a train might be far more convenient fordescribing things happening inside the train.

VELOCITY

So much for displacement. If that seems fairly simple, then the next logical question is how quickly wasSita moving? The quickness of someones displacement or the rate of change of their displacement is calledvelocity. The simplest kind of motion that an object can have is a uniform motion in a straight line. By uniformmotion is meant that in every second the body moves the same distance in the same direction as it did in everyother second. Every part of the body moves in exactly in the same way.

The speed of a moving body is the distance it moves per unit of time. If the speed is uniform, the objectmoves equal distances in each successive unit of time. Whether or not the speed is constant, the average speedis the distance the body moves, its displacement in space divided by the time required for the motion.

Average speed =elapsedtimetotal

travelledcedistotal

tani.e., v =

t

smeter / second.

Speed is thus a scalar quantity and hence the concept of speed does not include the idea of direction.The body moving with constant speed may move in a straight line or in a circle or in any one of an infinite varietyof paths so long as the distance moved in any unit of time so long as the distance moved in any unit of time is thesame as that moved in another equal unit of time.

The displacement covered by a body in one second is called the velocity It is a vector quantity. It isdirected along the direction of motion of body. The concept of velocity includes the idea of direction as well asmagnitude and velocity and is therefore a vector quantity and is the only physical quantity that gives direction of abody. ( velocity means direction aware )

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A car moving at constant speed along a winding road has a changing velocity because the direction ofmotion is changing.

Velocity is therefore defined as the time rate of change of displacement.

Average velocity = total displacement / total time elapsed i.e., v = s / t meter / second.

Constant velocity is a particular case of constant speed. Not only does the distance traveled per unit time

remain the same, but the direction as well does not change.

The statement A car is moving with a velocity of 20 kmph is incorrect because it is incomplete, in asmuch as the direction of motion must be stated in order to specify a velocity. For this reason one should alwaysuse the word speed when the direction of motion is not specified or when the direction is changing.

The unit of speed is same as that of the speed of a body. The magnitude of the velocity is equal to the speed of a body moving along a straight line. The speed the magnitude of velocity. The velocity of a body is uniform if both the magnitude and direction do not change.

If a body travels with a speed V1 for the first half of the journey time and with a speed V2 for the secondhalf of the journey time, then average velocity = ( V1 + V2 ) / 2

If a body covers first half of its journey with a uniform speed V1 and the second half of its journey with auniform speed V2 then the average speed for the whole journey is 2 V1V2 / ( V1 + V2 ).

We are frequently interested in knowing the speed and velocity of a body at a particular instant and notmerely the average value over a considerable time interval. For this purpose it is convenient to consider theratio s / t, where s is the change of displacement which the body has during the small time interval t.If t is made smaller and smaller, approaching zero but never reaching it, s becomes equally infinitesimaland the instantaneous velocity is the lowest possible value or limit of this ratio i.e., it is ds / dt. It is directedalong the tangent at that point if the path of the body is a curve.

If a body travels with velocity V1 for t1 seconds and with velocity V2 for t2 seconds then average velocity is( V1t1 + V2t2 ) / ( t1 + t2 )

If a body travels a distance s1 with velocity V1 and s2 with velocity V2 in the same direction then averagevelocity is ( s1 + s2 ) V1V2 / [ s1V2 + s2V1 ].

For average velocity we consider the duration of time of the displacement as well as the displacement itself.xf - xo

In Sitas case, her average velocity would be calculated as follows, if her displacement took 5 seconds.

Sometimes the information is given in a straight forward manner, as we just did, or it may be presentedgraphically. The principle behind finding the average velocity remains pretty much the same--change indisplacement over change in time.

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PHYSICS MASTER IIT ACADEMY, Vijayawada Study MaterialCase 1: Determine the average velocity from t = 3s to t = 7s, using thegraph shown below.

Case 2: Determine the average velocity for the particle according tothe graph shown below.

As long as the line is straight, pick any two points on the line. Ill

choose (1 s, 8 m) and (6 s, 3 m). Now you can plug them in, butremember that the displacement values are the y coordinates and timevalues are the x coordinates.

Ill explain what I mean by instantaneous velocity. This is the velocity that something is traveling at any pointin time. Average velocity occurs over a period of time. Instantaneous velocity might be found after exactly5.0124 seconds have elapsed.

Well, if you look at the graph near the top of this page, youll see that the slope of this graph is constant,since the line is straight. If the slope of this line is constant that means that the velocity at any point of timebetween zero and 9 seconds is equal to the slope of the line.

However, how would this work with the graph from the previous example? That line has anything but aconstant slope. Using the method of calculating average velocity will do no good whatsoever in assisting usin finding the instantaneous velocity. Instead, we will resort to using two other methods of determininginstantaneous velocity: the tangent line method and the calculus method.

For the first of the graphical example problems, the instantaneous velocity at 3 seconds could be best byusing the tangent line approach. Withthis method, you would:

A. Draw in a line tangent to the curve.B. Pick two points on that line. (3 s, 1 m) & (9 s, 2 m)C. Find the velocity by computing the slope.

The other means of computing instantaneous velocity uses calculus. Im not going to derive the formula for

the derivative, but you ought to remember that the whole concept of the derivative was conceived by IsaacNewton, because he needed to find the slope of a line at a point. Therefore, the equation definition ofinstantaneous velocity looks like:

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When does this method of determining velocity come into play? Basically, whenever you have an equation forposition and are asked to find the velocity--either in general or at some point.

Let's say that you want to determine a general equation for instantaneous velocity from a displacementfunction, then you would follow the example shown below.

Example 1: Determine the velocity of a particle at any point in time whose position is described asx(t) = 3 - 4t + 6t2.

Sometimes you want to get a specific value for the instantaneous velocity at some point in time. Youwould follow the same procedure as shown in the previous example. However, you would continue the analysis byevaluating the expression for a particular point in time. See the example below:

Example 2: Determine the velocity of a particle after 2 seconds whose position is described as x(t) = 3 - 4t + 6t2.

ACCELERATION

Well, we started looking at velocity because we understood that positions change and they do so oversome time period. The next topic, acceleration, arises for similar reasons: velocity can change and will do so overa period of time. In fact, this takes us back to our cause-effect relationships. Whenever unbalanced forces act onan object, they cause acceleration, which causes changes in velocity. This leads us to the kinematic definition ofaverage acceleration:

We can use this equation to compute average acceleration when given the appropriate information:

A ball hit the street rolling at 1.5 m/s and was seen 20 s later rolling down the bridge at 2.25 m/s. What was theaverage acceleration of the ball ?

The change in velocity may be due to the change in either magnitude or in direction, or in both magnitude anddirection.

If the direction of acceleration is parallel to the direction of motion, only the speedchanges. If the acceleration is at right angles to the direction of motion, only the direction changes. Acceleration in any other direction produces changes in both speed and direction.

Although we use the word speedto describe the magnitude of a velocity, there is no other correspondingword for the magnitude of an acceleration. Hence the term acceleration is used to denote either the vectorquantity or its magnitude.

The term acceleration will be used to refer to the instantaneous and not the average value unlessotherwise stated.

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If a body moves along a straight line with increasing velocity the acceleration is directed in the direction ofmotion of the body then the acceleration ispositive ( v > u ).

If a body moves along a straight line with decreasing velocity the acceleration is directed in the directionopposite to that of motion of the body then the acceleration is negative.

Negative acceleration is called retardation or deceleration. ( v< u )

When a body moves along a circular path with constant speed its velocity continually changes. This

change is due to the change in direction of velocity. This change in velocity in unit time is called normalacceleration. The normal acceleration is always perpendicular to the velocity. The acceleration and velocity of a body need not be in the same direction. Eg: a body thrown upwards For a freely falling body, the velocity changes in magnitude, hence it has acceleration. For a body moving round a circular path with a uniform speed, the velocity changes indirection, hence it

has acceleration. For a projectile, whose trajectory is a parabola, the velocity changes in direction and hence it has

acceleration.

When a body is moving with uniform acceleration, the average velocity is equal to the arithmetic mean of

initial and final velocities. i.e., Average velocity = ( v + u ) / 2 m / s .

If a body travels such that its displacement S t then it moves with uniform velocity and zero acceleration.

If a body travels such that its displacement S t2 then it moves with uniform acceleration and variablevelocity.

If a body has uniform speed, it may have an acceleration or zero acceleration, its velocity may be uniform. If a body has uniform velocity, it has no acceleration, its speed is uniform. If the acceleration of a body is zero, the body may be at rest or may be moving with uniform velocity. The body may have velocity without acceleration. ( Uniform velocity ) The body having zero velocity may possess uniform acceleration or non-uniform acceleration.

If a body moving with uniform acceleration has velocities u and v at two points in its path, the velocity atthe mid point is [(u2+v2) / 2]

However, acceleration like velocity may be interpreted from graphical information. Only the graphs examinedwhen we want to find the acceleration will plot velocity vs. time as opposed to the position versus time graphsused to interpret velocity graphically.

Case 1: Determine the average acceleration from t = 3s to t = 7s for an objecttraveling as described by the graph below.

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PHYSICS MASTER IIT ACADEMY, Vijayawada Study MaterialCase 2: Determine the average acceleration for the particle according to thegraph shown below.

As long as the line is straight, pick any two points on the line. Ill choose (4 s, 6

m/s) and (8 s, 4 m/s). Now you can plug them in, but remember that the velocityvalues are the y coordinates and time values are the x coordinates.

Hopefully, you are beginning to see the similarities between the relationship between velocity anddisplacement and the relationship between acceleration and velocity.

Just as before, there are two ways to compute the instantaneous acceleration. The tangent methodapplies when attempting to ascertain the acceleration from a graph. The derivative method would be used todetermine the acceleration from an equation. Since the tangent method is essentially the same as before, Iwont give an example of that type of problem. However, I will take a moment to discuss the derivativemethod to find acceleration.

Once again the similarities with the previous relationship are uncanny, almost eerie. The equationdefinition of instantaneous acceleration is:

This method of determining acceleration is used whenever you have an equation for position or an equationfor velocity. Why either of these? Since you know that velocity is the derivative of position, being given aposition equation just means that you have one more derivative to take before you get to analyze theacceleration.

Example: Determine the acceleration of a particle at any point in time whose position is described as

x(t) = 3 - 4t + 6t2 + 2t3.

First, find the expression for velocity.

Now you are ready to find the acceleration.

It is true that acceleration can vary with time, but the extent of kinematic analysis stops with accelerationregardless of whether acceleration is varying with time.

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DISPLACEMENT "Distance between two points in a particular direction is called Displacement."

ORDisplacement may also be defined as "the minimum distance between two points in a particular direction."

It is a vector quantity and is always directed from the initial point to the terminal point.

It is denoted by "d".SPEED

"Distance covered by a moving body in one second is called its Speed".OR

"Distance covered by a body in unit time is called Speed".

Speed is a scalar quantity.FORMULA

Speed = Distance traveled/Time takenOR

v = S/t

UNIT Unit of speed in S.I system is "m/sec".

VELOCITY "Distance covered by a body in a particular direction in one second is called Velocity".

OR"Displacement of a body in unit time is called Velocity".

OR"Change of position of a body per second in a particular direction is called Velocity."

FORMULA velocity = displacement/time

UNIT In S.I system unit of velocity is meter/second.It is a vector quantity.

ACCELERATION "The rate of change of velocity of a body is called Acceleration."

OR"Change in velocity of a body in unit time is called its acceleration."It is denoted by "a".It is a vector quantity.If a body moves with uniform velocity or constant velocity then its acceleration will be zero.UNIT: m/sec2.

FORMULA Acceleration = change in velocity/time

ORa = V/t

Graphs of Motion; Velocity

Motion with constant velocity

In example m, an object is moving at constant speed in one direction. We can tell this because every twoseconds, its position changes by five meters. In algebra notation, we'd say that the graph of x vs. t shows thesame change in position, x = 5.0 m, over each interval of t = 2.0 s. The object's velocity or speed is obtained bycalculating v = x/t = (5.0 m)/(2.0 s) = 2.5 m/s. In graphical terms, the velocity can be interpreted as the slope ofthe line. Since the graph is a straight line, it wouldn't have mattered if we'd taken a longer time interval andcalculated v = x/t = (10.0 m)/(4.0 s). The answer would still have been the same, 2.5 m/s.

Note that when we divide a number that has units of meters by another number that has units of seconds,

we get units of meters per second, which can be written m/s. This is another case where we treat units as if theywere algebra symbols, even though they're not.

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m) Motion with constant velocity.n) Motion that decreases x isrepresented with negative values ofx and v.

o) Motion with changing velocity.

In example n, the object is moving in the opposite direction: as time progresses, its x coordinatedecreases. Recalling the definition of the notation as "after minus before," we find that t is still positive, but xmust be negative. The slope of the line is therefore negative, and we say that the object has a negative velocity, v= x/t = (-5.0 m)/(2.0 s) = -2.5 m/s. We've already seen that the plus and minus signs of x values have theinterpretation of telling us which direction the object moved. Since t is always positive, dividing by t doesn't

change the plus or minus sign, and the plus and minus signs of velocities are to be interpreted in the same way. Ingraphical terms, a positive slope characterizes a line that goes up as we go to the right, and a negative slope tellsus that the line went down as we went to the right.

Motion with changing velocity

Now what about a graph like figure o? This might be a graph of a car's motion as the driver cruises downthe freeway, then slows down to look at a car crash by the side of the road, and then speeds up again,disappointed that there is nothing dramatic going on such as flames or babies trapped in their car seats. (Notethat we are still talking about one-dimensional motion. Just because the graph is curvy doesn't mean that the car'spath is curvy. The graph is not like a map, and the horizontal direction of the graph represents the passing of time,not distance.)

Example o is similar to example m in that the object moves a total of 25.0 m in a period of 10.0 s, but it isno longer true that it makes the same amount of progress every second. There is no way to characterize theentire graph by a certain velocity or slope, because the velocity is different at every moment. It would be incorrectto say that because the car covered 25.0 m in 10.0 s, its velocity was 2.5 m/s. It moved faster than that at thebeginning and end, but slower in the middle. There may have been certain instants at which the car was indeedgoing 2.5 m/s, but the speedometer swept past that value without "sticking," just as it swung through various othervalues of speed. (I definitely want my next car to have a speedometer calibrated in m/s and showing both negativeand positive values.)

We assume that our speedometer tells us what is happening to the speed of our car at every instant, buthow can we define speed mathematically in a case like this? We can't just define it as the slope of the curvygraph, because a curve doesn't have a single well-defined slope as does a line. A mathematical definition thatcorresponded to the speedometer reading would have to be one that attached a different velocity value to a singlepoint on the curve, i.e., a single instant in time, rather than to the entire graph. If we wish to define the speed atone instant such as the one marked with a dot, the best way to proceed is illustrated in p, where we have drawnthe line through that point called the tangent line, the line that "hugs the curve." We can then adopt the followingdefinition of velocity:

One interpretation of this definition is that the velocity tells us how many meters the object would havetraveled in one second, if it had continued moving at the same speed for at least one second. To some people thegraphical nature of this definition seems "inaccurate" or "not mathematical." The equation by itself, however, isonly valid if the velocity is constant, and so cannot serve as a general definition.

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p / The velocity at any givenmoment is defined as the slope ofthe tangent line through therelevant point on the graph.

q / Example: finding the velocity atthe point indicated with the dot.

r / Reversing the direction ofmotion.

Conventions about graphing

The placement of t on the horizontal axis and x on the upright axis may seem like an arbitrary convention,or may even have disturbed you, since your algebra teacher always told you that x goes on the horizontal axisand y goes on the upright axis. There is a reason for doing it this way, however. In example q, we have an objectthat reverses its direction of motion twice. It can only be in one place at any given time, but there can be morethan one time when it is at a given place. For instance, this object passed through x = 17 m on three separateoccasions, but there is no way it could have been in more than one place at t = 5.0 s. Resurrecting someterminology you learned in your trigonometry course, we say that x is a function of t, but t is not a function of x. Insituations such as this, there is a useful convention that the graph should be oriented so that any vertical linepasses through the curve at only one point. Putting the x axis across the page and t upright would have violatedthis convention. To people who are used to interpreting graphs, a graph that violates this convention is asannoying as fingernails scratching on a chalkboard. We say that this is a graph of "x versus t." If the axes were theother way around, it would be a graph of "t versus x." I remember the "versus" terminology by visualizing thelabels on the x and t axes and remembering that when you read, you go from left to right and from top to bottom.

Discussion Questions

A Ram is running slowly in gym class, but then he notices Shyam watching him, so he speeds up to try toimpress him. Which of the graphs could represent his motion?

B The figure shows a sequence of positions for two racing tractors. Compare the tractors' velocities as the raceprogresses. When do they have the same velocity?

C If an object had a straight-line motion graph with x=0 and t 0, what would be true about its velocity? Whatwould this look like on a graph? What about t=0 and x 0?

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D If an object has a wavy motion graph like the one in example (e) on the previous page, which are the points atwhich the object reverses its direction? What is true about the object's velocity at these points?

E Discuss anything unusual about the following three graphs.

F I have been using the term "velocity" and avoiding the more common English word "speed," becauseintroductory physics texts typically define them to mean different things. They use the word "speed," and thesymbol "s" to mean the absolute value of the velocity, s = |v|. Although I've chosen not to emphasize thisdistinction in technical vocabulary, there are clearly two different concepts here. Can you think of an exampleof a graph of x-versus-t in which the object has constant speed, but not constant velocity?

G

For the graph shown in the figure, describe how the object's velocity changes.

H Two physicists duck out of a boring scientific conference to go get some coffee. On the way to the cafe, theywitness an accident in which a pedestrian is injured by a hit-and-run driver. A criminal trial results, and theymust testify. In her testimony, Dr. Sudhakar says, "The car was moving along pretty fast, I'd say the velocitywas + 40 km/hr. They saw the old lady too late, and even though they slammed on the brakes they still hit herbefore they stopped. Then they made a U turn and headed off at a velocity of about -20 km/hr, I'd say." Dr.Azeem says, "He was really going too fast, maybe his velocity was -35 or -40 km/hr. After he hit Mrs. Seema,he turned around and left at a velocity of, oh, I'd guess maybe +20 or +25 km/hr." Is their testimonycontradictory? Explain.

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p Week II Kinematics Viii

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