1. Acceleration

2. Acceleration Any change in velocity
a = change in velocity/change in time
a = v/ t = m/s2
Which car has the greatest acceleration?

3. Types of Acceleration
Example: A car is traveling at 1m/s when it reaches the edge of town. It then speeds up to 5 m/s in 20 seconds. What is its average acceleration?
a = v/t
a = 5m/s – 1m/s
20s
=4m/s/20s = 0.20m/s2

4. Graphing Acceleration
When an accelerating object’s motion is graphed on a position/time graph, the graph will be curved.
On a velocity/time graph, the slope of the line will be the objects acceleration.
To show acceleration, graph velocity on the y axis and time on the x-axis

5. Types of Acceleration Average Instantaneous acceleration
acceleration is over a length of time.
Instantaneous acceleration
Acceleration at an given instant in time.

6. Constant Acceleration
Objects with constant acceleration
Vf = Vi + at
Example: If a car with a velocity of 2.0 m/s at t = 0 accelerates a rate of +4.0 m/s2 for 2.5s, what is its velocity at t = 2.5s?
Known?
Vi = 2.0m/s a = +4.0m/s2 t = 2.5s
Unknown= Vf
Vf = 2.0m/s + (4.0m/s2)(2.5s)
Vf = 12 m/s

7. Acceleration Due to Gravity
Near Earth’s Surface, all objects fall toward the center of the Earth with an acceleration of
9.80m/s2
Ignoring air resistance.
Free Falls – objects fall with the acceleration due to gravity
Example: A ball is dropped from the roof and allowed to fall for 30s. What is the velocity of the ball when it hits the ground?
Vf = Vi + at
Vf­ = 0 + (9.80m/s2)(3 0m)
Vf = 294m/s

8. Warm Up
If a ball is dropped off a cliff and lands with a velocity of 35m/s, how long was it falling?

9. Warm Up
A penny is dropped off the Empire State Building, 381m tall. With what velocity will it hit the sidewalk?

10. Constant Acceleration
Remember…. a = V/t
Using that formula, we can find many other pieces of information
During Constant Acceleration:
Velocity can be found by:
Vf = Vi + at

11. Constant Acceleration
Displacement can be found if:
Time and Velocity are known
d = 0.5(Vf +Vi)t
Example: What is the displacement of a train as it is accelerated uniformly from +11m/s to +33m/s in a 20.0s interval?
d = 0.5(Vf + Vi)t
d = 0.5(33m/s + 11m/s)20.0s
d = +4.4 x 102m

12. Checkpoint
How long does it take a car, starting from rest and accelerating at a rate of 2.5m/s2 to reach a velocity of 25m/s?

13. Constant Acceleration (finding displacement
Acceleration and Time are Known
d = Vit + 0.5at2
Because:
Vf = Vi + at and d = 0.5(Vf + Vi)t
We can sub the first equation for Vf in the second.
So, d = 0.5((Vi + at) +Vi)t
d = 0.5(2Vi + at)t
d = Vit + at2

14. Constant Acceleration
Example:
A car starting from rest accelerates uniformly at +6.1m/s2 for 7.0 seconds. How far does the car move?
Given: Vi = 0 a = +6.1m/s2 t = 7.0s
d = Vit + 0.5at2
d = (0)(7.0s) + 0.5(6.1m/s2)(7.0s)2
d = m
d= m

15. Constant Acceleration
Velocity and Acceleration are Known
Vf2 = Vi2 + 2ad
Because
Vf = Vi + at and d = 0.5(Vf + Vi)t
Solve first equation for t
t = (Vf - Vi)/a
Sub t into the second equation
d = 0.5(Vf + Vi)(Vf – Vi)/a
or d = (0.5(Vf – Vi)2 )/a

16. Constant Acceleration
Example: An airplane must reach a velocity of 71m/s for takeoff. If the runway is 1.0km long, what must the constant acceleration be?
Given: Vi = 0m/s Vf = 71m/s
d = +1.0km = +1.0 x 103m
Unknown: acceleration
Vf2 = Vi2 +2ad
712m/s = 0.0m/s + 2(1.0 x 103)a
Solve for a
712 m/s = 2(1.0 x 103)a
(712m/s)/((2)1000m) = a
a = 2.52m/s2

17. Constant Acceleration
OVERALL, the four kinematics (motion for uniform acceleration) equations are:
Determine the variables you know and want to know
Determine the necessary equation Vf Vi d a t
Vf = Vi + at x
d = 0.5(Vf + Vi)t
d = Vit + 0.5at2
Vf2 = Vi2 + 2ad

18. Example Which equation? Vf2 = Vi2 + 2ad 02 = 552 + 2(-11)d
A car is traveling at 55 m/s when it slows with an acceleration of -11 m/s2. How far does it take to stop?
Known?
Vi = 55m/s
a = -11m/s2
Vf = 0m/s
Unknown?
D = ?
Which equation?
Vf2 = Vi2 + 2ad
02 = (-11)d
0 = d
-3025 = -22d
137.5m = d

19. Example 2 If you drop a golf ball, how far does it fall in 0.5 s?
Known?
Vi = 0m/s
t = 0.5s
a = 9.8m/s2
Unknown?
d = ?
Which equation?
d = Vit + 0.5at2
d = 0(d) + 0.5(9.8)0.52
d = (0.25)
d = 1.225m

20. Example 3 Which equation? Vf = Vi + at 66 = 0 + a(3.0) 22m/s2 = a
A steam driven catapult accelerates a 20 ton aircraft from 0 to 66 m/s in 3.0 s to launch the plane from the deck of an aircraft carrier. Find the rate of acceleration.
Known?
Vi = 0
Vf = 66m/s
T = 3.0s
Unknown?
a = ?
Which equation?
Vf = Vi + at
66 = 0 + a(3.0)
22m/s2 = a

21. Warm Up
A penny is dropped off the Empire State Building, 350m high. How fast is it going when it hits the ground?