1. 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

2. 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

3. 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?

4. Average speed and velocity
Average speed  distance traveled divided by the total elapsed time
SI units, meters/second (m/s)
Scalar quantity
Always positive

5. Average speed and velocity
Average velocity  displacement divided by the total elapsed time
SI units of m/s
Vector quantity
Can be positive or negative

6. 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?

7. 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

8. 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.

9. Acceleration
Acceleration – how fast you speed up, slow down, or change direction; it’s the rate at which velocity changes. Two examples:
t (s)
v (mph) 55 1 57 2 59 3 61
t (s)
v (m/s) 34 1 31 2 28 3 25
a = -3
m / s s
= -3 m / s 2
a = +2 mph / s

10. 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

11. 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

12. Acceleration due to Gravity
Near the surface of the Earth, all objects accelerate at the same rate (ignoring air resistance).
This acceleration vector is the same on the way up, at the top, and on the way down!
a = -g = m/s2
9.81 m/s2
Interpretation: Velocity decreases by 9.81 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

13. 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?

14. 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 that
this equation is dimen-
sionally correct?
v  t

15. Velocity & Acceleration Sign Chart+ -
Moving forward; Speeding up
Moving backward; Slowing down
Moving forward; Slowing down
Moving backward; Speeding up

16. Acceleration Under which scenarios does the car’s speed
increase? Decrease?

17. Kinematics Formula Summary
For 1-D motion with constant acceleration:
vf = v0 + a t
vavg = (v0 + vf ) / 2
x = v0 t + ½ a t 2
vf2 – v02 = 2 a x
(derivations to follow)

18. Kinematics Derivations
a = v / t (by definition)
a = (vf – v0) / t
 vf = v0 + a t
vavg = (v0 + vf ) / 2 will be proven when we do graphing.
x = v t = ½ (v0 + vf) t = ½ (v0 + v0 + a t) t  x = v0 t + a t 2
(cont.)

19. Kinematics Derivations (cont.)
vf = v0 + a t  t = (vf – v0) / a x = v0 t a t 2  x = v0 [(vf – v0) / a] + a [(vf – v0) / a]  vf2 – v02 = 2 a x
Note that the top equation is solved for t and that expression for t is substituted twice (in red) into the x equation. You should work out the algebra to prove the final result on the last line.

20. Sample Problems
You’re riding a unicorn at 25 m/s and come to a uniform stop at a red light 20 m away. What’s your acceleration?
-15.6 m/s2