Motion in a straight line

© 2013 Pearson Education, Inc.

.Slide 2-2

Kinematics is the study of motion without going into its causes.

This chapter deals with motion along a straight line, i.e. rectilinear motion.

The motion is the change in position of an object with respect to time.

Slide 2-3


The distance an object travels is a scalar quantity, independent of direction.

The displacement of an object is a vector quantity, equal to the final position minus the initial position.

An object’s speed v is scalar quantity, independent of direction.

Speed is how fast an object is going; it is always positive.

Velocity is a vector quantity that includes direction.

In one dimension the direction of velocity is specified by the or sign. Slide 2-28


Distance = length of the actual path taken to go from source to destination

Displacement = length of the straight line joining the source to the destination or in other words the length of the shortest path


Example: If a boy walks from B to D [arc] in a circular path, the distance will be the semicircle of the circle, while the displacement will be the diameter BD.

B

A

CD


Rohit and Seema both start from their house. Rohit walks 2 km to the east while Seema walk 1 km to the west and then turns back and walks 1 km.

Seema is back home and her displacement is 0 m.

This is because direction of motion is different in both cases.

You require both distance and direction to determine displacement.

Distance travelled by them is the same (2 km)


Uniform motion is when equal displacements occur during any successive equal-time intervals.

Uniform motion is always along a straight line.

Eg; While driving a car at a perfectly steady 60 kmph, this means there is a change in the position by 60 km for every time interval of 1 hour.

Slide 2-20

Riding steadily over level ground is a good example of uniform motion.


An object’s motion is uniform if and only if its position-versus-time graph is a straight line.

The average velocity is the slope of the position-versus-time graph.

The SI units of velocity are m/s.

Slide 2-21



25

20

15

10

5

010 20 30 40 500

Time (s)

Distance – Time graph


1.25

1.0

0.75

0.5

0.25

010 20 30 40 500

Time (s)

Speed – Time graph


1.25

1.0

0.75

0.5

0.25

010 20 30 40 500

Time (s)

Velocity – Time graphNon-uniform Motion

Acceleration = 0.125 m/s2


An object that is speeding up or slowing down is not in uniform motion.

In this case, the position-versus-time graph is not a straight line.

We can determine the average speed Vav between any two times separated by time interval t by finding the slope of the straight-line connection between the two points.

The instantaneous velocity is the object’s velocity at a single instant of time t.

The average velocity Vav s/t becomes a better and better approximation to the instantaneous velocity as t gets smaller and smaller.

Slide 2-31

© 2013 Pearson Education, Inc. Slide 2-32

Motion diagrams and position graphs of an accelerating rocket.


As ∆t continues to get smaller, the average velocity vavg ∆s/∆t reaches a constant or limiting value.

The instantaneous velocity at time t is the average velocity during a time interval ∆t centered on t, as ∆t approaches zero.

In calculus, this is called the derivative of s with respect to t.

Graphically, ∆s/∆t is the slope of a straight line. In the limit ∆t 0, the straight line is tangent to the curve. The instantaneous velocity at time t is the slope of the line

that is tangent to the position-versus-time graph at time t.

Slide 2-33


ds/dt is called the derivative of s with respect to t.

ds/dt is the slope of the line that is tangent to the position-versus-time graph.

Consider a function u that depends on time as u ctn, where c and n are constants:

The derivative of a constant is zero:

The derivative of a sum is the sum of the derivatives. If u and w are two separate functions of time, then:

Slide 2-46


Taking the derivative of a function is equivalent to finding the slope of a graph of the function.

Similarly, evaluating an integral is equivalent to finding the area under a graph of the function.

Consider a function u that depends on time as u ctn, where c and n are constants:

The vertical bar in the third step means the integral evaluated at tf minus the integral evaluated at ti.

The integral of a sum is the sum of the integrals. If u and w are two separate functions of time, then:

Slide 2-60


The SI units of acceleration are (m/s)/s, or m/s2. It is the rate of change of velocity and measures how

quickly or slowly an object’s velocity changes. The average acceleration during a time interval ∆t

is:

Graphically, aavg is the slope of a straight-line velocity-versus-time graph.

If acceleration is constant, the acceleration as is the same as aavg.

Acceleration, like velocity, is a vector quantity and has both magnitude and direction.

Slide 2-64


1.25

1.0

0.75

0.5

0.25

010 20 30 40 500

Time (s)

Velocity – Time graphUniform Acceleration

Acceleration = 0.125 m/s2


1.25

1.0

0.75

0.5

0.25

010 20 30 40 500

Time (s)

Velocity – Time graphNon-uniform Acceleration


v

u

0

t0

Time (s)


Initial velocity = uFinal velocity = vTime = tAcceleration = aDisplacement = s


v

u

0t0

Time (s)




v

u

0t0

Time (s)




Derivation of equation 1; atvv 0

dtdva a is constant adtdv

Integrate both sides with respect to time from 0 to t

atvvatv

dtaadtvdt

vv

v t t

0

0 0 0

0][


t

0 00 0

2

0

Derivation of eqs.2

( )

Integrate both sides with respect to time from 0 to t

dx

2

o o

o

t tt t

o

o o

o

dxv dx vdt v v at dx v at dtdt

dx v dt atdt

v

atx x

dt a tdt v

t

t

v

x

2

0

2 2

2

( 0) 0 2 2

t

o o o

ta

t tx x v t a v t a

2

2o oatx x v t

x x


a = dv/dt=dv/dx × dx/dt a =vdv/dx vdv= adx x x

∫u vdv = ∫x adxv 2 - u2 = a ( x- x0 ) =as───── 2v2 – u2 = 2as


The motion of an object moving under the influence of gravity only, and no other forces, is called free fall.

Two objects dropped from the same height will, if air resistance can be neglected, hit the ground at the same time and with the same speed.

Consequently, any two objects in free fall, regardless of their mass, have the same acceleration:

Slide 2-94

In the absence of air resistance, any twoobjects fall at the same rate and hit theground at the same time. The apple andfeather seen here are falling in a vacuum.


Figure (a) shows the motion diagram of an object that was released from rest and falls freely.

Figure (b) shows the object’s velocity graph.

The velocity graph is a straight line with a slope:

Where g is a positive number which is equal to 9.80 m/s2 on the surface of the earth.

Other planets have different values of g.

Slide 2-95


When two bodies are moving in the same direction parallel to each other;

vab = va – vb

When two bodies are moving in opposite directions;

vab = va + vb

When two bodies make an angle with each other

vab = \/va2 + vb2 + 2vabcos


The bus moved away from the treeThe person is comparing the position of the bus with respect to the position of the treeReference (or origin) is position of the tree

0 5 10


The tree moved away from the bus.

The person is comparing the position of the tree with respect to the position of the bus.

Reference (or origin) is position of the bus.

0510


Both the observations are correct. The difference is what is taken as the origin.Motion is always relative. When one says that a object is moving, he/she is comparing the position of that object with another object.Motion is therefore change in position of an object with respect to another object over time.Kinematics studies motion without delving into what caused the motion.

Motion in a straight line

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