This document discusses acceleration and kinematic equations for objects experiencing constant acceleration. It defines acceleration as the rate of change of velocity and introduces the average acceleration equation. It then derives and explains the equations for displacement given constant acceleration using the relationships between velocity, acceleration, displacement and time. Sample problems demonstrate applying the equations to calculate values when given acceleration, initial/final velocities, displacement or time. The key equations covered are for average and final velocity, and displacement given constant acceleration.

1. Acceleration
p. 48-58

2. Objectives
The student will be able to:
• Describe motion in terms of changing velocity
• Compare graphical representations of accelerated
and non-accelerated motions
• Apply kinematic equations to calculate distance,
time, or velocity under conditions of constant
acceleration

3. Acceleration
Acceleration: the rate of change of velocity
• Units: meters per second per second or meters per
second squared (m/s2)
• Symbol: a

4. Average Acceleration Equation
𝑎 =
∆𝑣
∆𝑡
=
𝑣 𝑓 − 𝑣 𝑜
𝑡𝑓 − 𝑡 𝑜
average acceleration =
change in velocity
time required for change
• Acceleration has both direction and magnitude.

5. Sample Problem 2B – p.49
A shuttle bus slows down with an average acceleration of
-1.8 m/s2.
How long does it take the bus to slow from 9.0 m/s to a
complete stop?

6. Problem Tip
Watch for implied data in problem statements, such as
“starts at rest” (vi = 0 m/s) or “comes to rest” (vf = 0 m/s)

7. Practice 2B Problems
p.49 #1, 3-5

8. The signs of the velocity and acceleration combine to
describe an object’s motion.
𝒗𝒊 a Motion
+ + speeding up
_ _ speeding up
+ _ slowing down
_ + slowing down
- or + 0 constant velocity
0 - or + speeding up from rest
0 0 remaining at rest

9. Graphing Acceleration
On a velocity versus time graph, the slope of the line
connecting one point and the next indicates the average
acceleration.
slope =
change in vertical
change in horizontal
𝑎 𝑎𝑣𝑔 =
∆𝑣
∆𝑡

10. Acceleration Graph
Describe the acceleration at points A, B, C.

11. Think/Pair/Share
Conceptual Challenge on p.50

12. Constant Acceleration
What is it?
Demo - Walking

13. Deriving Displacement w/Constant Acceleration
What we know so far..
𝑣 𝑎𝑣𝑔 =
∆𝑥
∆𝑡
For objects moving with constant acceleration
𝑣 𝑎𝑣𝑔 =
𝑣 𝑖+𝑣 𝑓
2
 avg. velocity =
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦+𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
2

14. Deriving Displacement w/Constant Acceleration
∆𝑥
∆𝑡
= 𝑣 𝑎𝑣𝑔=
𝑣𝑖 + 𝑣 𝑓
2

15. Displacement w/Constant Acceleration
∆𝑥 =
1
2
(𝑣𝑖 + 𝑣 𝑓)∆𝑡
displacement =
1
2
(initial velocity + final velocity)(time interval)
• You will know that acceleration is constant by the
phrase “uniform negative/positive acceleration”

16. Sample Problem 2C– p.53
A race car reaches a speed of 42 m/s. It then begins a
uniform negative acceleration, using its parachute and
braking system, and comes to rest 5.5 s later.
Find out the distance the car travels during braking.

17. Practice 2C Problems
p.53 #2-4

18. Velocity with Constant Acceleration
What if we don’t know the vf but we still want to
calculate displacement…
𝑎 𝑎𝑣𝑔 =
∆𝑣
∆𝑡
=
𝑣 𝑓−𝑣 𝑖
∆𝑡

19. Velocity with Constant Acceleration
𝑣 𝑓 = 𝑣𝑖 + 𝑎∆𝑡
final velocity = initial velocity + (acceleration · time interval)

20. Displacement w/Constant Acceleration
We can combine the previous equations:
∆𝑥 =
1
2
(𝑣𝑖 + 𝑣 𝑓)∆𝑡 and 𝑣 𝑓 = 𝑣𝑖 + 𝑎∆𝑡
to form
∆𝑥 =
1
2
(𝑣𝑖 + 𝑣𝑖 + 𝑎∆𝑡)∆𝑡
∆𝑥 =
1
2
(2𝑣𝑖∆𝑡 + 𝑎∆𝑡2)

21. Displacement w/Constant Acceleration
∆𝑥 = 𝑣𝑖∆𝑡 +
1
2
𝑎∆𝑡2
displacement = (initial velocity · time interval) +
1
2
acceleration · (time interval)2

22. Sample Problem 2D – p.55
A plane starting at rest at one end of a runway undergoes
a uniform acceleration of 4.8 m/s2 for 15 s before takeoff.
What is its speed at takeoff?
How long must the runway be for the plane to be able to
take off?

23. Practice 2D Problems
p.55 #1-3

24. So far all equations have required knowing the time
interval.
If we don’t know t, we need to rearrange an equation
and use substitution.

25. Rearrange for ∆𝑡:
∆𝑥 =
1
2
(𝑣𝑖 + 𝑣 𝑓)∆𝑡

26. Then substitute into:
𝑣 𝑓 = 𝑣𝑖 + 𝑎∆𝑡

27. Final Velocity After any Displacement
𝑣 𝑓
2 = 𝑣𝑖
2 + 2𝑎∆𝑥
• When you use this equation, you must take the
square root of the equation to find the vf.
• The square root can be either positive or negative,
you will determine which value is right by
reasoning based on the direction of motion.

28. Sample Problem 2E – p.57
A person pushing a stroller starts from rest,
uniformly accelerating at a rate of 0.500 m/s2.
What is the velocity of the stroller after it has traveled
4.75 m?

29. Practice 2E Problems
p.58 #1,3,5

31. Homework
p.70-71 #18-25