Chapter 2 One Dimensional Kinematics Learning Objectives Position, Distance, and Displacement Average Speed and Velocity Instantaneous Velocity Acceleration.

Chapter 2 One Dimensional Kinematics

Learning Objectives

• Position, Distance, and Displacement• Average Speed and Velocity• Instantaneous Velocity• Acceleration• Motion with Constant Acceleration• Applications of the Equations of Motion• Freely Falling Objects

PowerPoint presentations are compiled from Walker 3rd Edition Instructor CD-ROM and Dr. Daniel Bullock’s own resources

Position, Distance, and Displacement

• Coordinate system defines position

• Distance total length of travel– (SI unit = meter, m)– Scalar quantity

• Displacement change in position– Change in position = final pos. – initial pos. x = xf – xi

– Vector quantity

Position, Distance, and Displacement

Before describing motion, you must set up a coordinate system – define an origin and a positive direction.

The distance is the total length of travel; if you drive from your house to the grocery store and back, what is the total distance you traveled?

Displacement is the change in position. If you drive from your house to the grocery store and then to your friend’s house, what is your total distance? What is your displacement?

Average speed and velocity

• Average speed distance traveled divided by the total elapsed time

– SI units, meters/second (m/s)– Scalar quantity– Always positive

timeelapsed

distance speed Average

• Is the average speed of the red car:A. 40 mi/h

B. More than 40 mi/h

C. Less than 40 mi/h

Average speed and velocity

Average speed and velocity

• Average velocity displacement divided by the total elapsed time

– SI units of m/s– Vector quantity– Can de positive or negative

if

ifav tt

xx

t

xv

timeelapsed

ntdisplaceme velocity Average

Average speed and velocity• Average velocity = displacement /

elapsed timeWhat’s your average velocity if you return to your starting point?

What if the runner sprints 50 m in 8 s?

What if he walks back to the starting line in 40 s?

Can you calculate: What is his average sprint velocity? His average walkingvelocity? And his average velocity for the entire trip?

Graphical Interpretation of Average Velocity

• The same motion, plotted one-dimensionally and as an position vs. time (x-t) graph:

Position vs time graphs give us information about:• average velocity slope of a line on a x-t plot is equal to the average velocity over that interval

12

12

run

rise slope

xx

yy

Graphical Interpretation of average velocity

What’s the average velocity between the intervals t = 0 s t = 3 s? Is the particle moving to the left or right?What’s the average velocity between the intervals t = 2 s t = 3 s? Is the particle moving to the left or right?

Instantaneous Velocity

• Instantaneous velocity

• This means that we evaluate the average velocity over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous velocity.

• Magnitude of the instantaneous velocity is known as the instantaneous speed

Instantaneous Velocity• As t smaller,

the ratio x/t becomes constant

• Consider the simple case of an object with constant velocity

• In this case as t gets smaller the ratio remains constant

Instantaneous velocity• If we have a more complex motion

• This plot shows the average velocity being measured over shorter and shorter intervals. The instantaneous velocity is tangent to the curve.

Instantaneous velocityIs the instantaneous

velocity at t = 0.5 s

A. Greater thanB. Less thanC.Or equal to the

instantaneous velocity at t = 1.0 s

Instantaneous velocityGraphical Interpretation of Average and

Instantaneous Velocity

Average velocity is the slope of the straight line connecting two points corresponding to a given time interval

Instantaneous velocity is the slope of the tangent line at a given instant of time

Acceleration• Average acceleration the change in

velocity divided by the time it took to change the velocity

– SI units meters/(second · second), m/s2

– Vector quantity– Can be positive or negative– Accelerations give rise to force

Acceleration• What does it mean to

have an acceleration of 10 m/s2 ?

Time (s) Velocity (m/s)

0 0

1 10

2 20

3 30

Acceleration• Instantaneous acceleration - This means that

we evaluate the average acceleration over a shorter and shorter period of time; as that time becomes infinitesimally small, we have the instantaneous acceleration.

• When acceleration is constant, the instantaneous and average accelerations are equal

t

v

0t

lim a

Graphical interpretation of Acceleration

• Velocity vs time (v-t) graphs give us information about: average acceleration, instantaneous acceleration

• average velocity slope of a line on a x-t plot is equal to the average velocity over that interval

• the “+” 0.25 m/s2 means the particle’s speed is increasing by 0.25 m/s every second• What does the “-” 0.5 m/s2 mean?

Graphical interpretation of Acceleration

Acceleration• Acceleration (increasing speed) and deceleration

(decreasing speed) should not be confused with the directions of velocity and acceleration:

• In 1-D velocities & accelerations can be “+” or “-” depending on whether they point in the “+” or “-” direction of the coordinate system

• Leads to two conclusion– When the velocity & acceleration have the same sign

the speed of the object increases (in this case the velocity & acceleration point in the same direction)

– When the velocity & acceleration have opposite signs, the speed of the object decreases (in this case the velocity & acceleration point in opposite directions

AccelerationUnder which scenarios does the car’s speed increase? Decrease?

Motion with constant acceleration

• If the acceleration is constant, the velocity changes linearly:

• Average velocity:• v0 = initial velocity• a = acceleration• t = time• Can you show thatthis equation is dimen-sionally correct?• v t

Motion with constant acceleration

• Average velocity:

• Position as a function of time:

• Velocity as a function of position:

Motion with constant acceleration• The relationship between position and

time follows a characteristic curve.

• x t2

• If the time doubles what happens to the position?

Motion with Constant Acceleration

Can you derive these equations?

Motion with constant accelerationA park ranger driving on a back country road suddenly sees a deer

“frozen” in the headlights. The ranger who is driving at 11.4 m/s,

immediately applies the brakes and slows with an acceleration of 3.8

m/s2. If the deer is 20 m from the ranger’s vehicle when the brakes are

applied, how close does the ranger come to hitting the deer? How

much time is needed for the ranger’s vehicle to stop?

Motion with constant accelerationDoes the velocity vary uniformly with distance?

Motion with constant accelerationFree fall is the motion of an object subject only to

the influence of gravity. The acceleration due to gravity is a constant, g.

Motion with constant accelerationAn object falling in air is subject to air resistance (and therefore is not freely falling).

• Free fall is the motion of an object subject only to the influences of gravity• An object is in free fall as soon as it is released

Free falling objects

Free fall from rest

Trajectory of a projectile

Chapter 2 summary

• Distance: total length of travel

• Displacement: change in position

• Average speed: distance / time

• Average velocity: displacement / time

• Instantaneous velocity: average velocity measured over an infinitesimally small time

Chapter 2 summary

• Instantaneous acceleration: average acceleration measured over an infinitesimally small time

• Average acceleration: change in velocity divided by change in time

• Deceleration: velocity and acceleration have opposite signs

• Constant acceleration: equations of motion relate position, velocity, acceleration, and time

• Freely falling objects: constant acceleration g = 9.81 m/s2