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Acceleration
Acceleration: The rate at which the velocity changes during a given amount of time. acceleration = change in velocity or a = Δv = v f
–v i
elapsed time Δt t f
– t i where the terms v f
and v i
mean final velocity and initial velocity, respectively.
The SI unit for acceleration is the meter per second squared (m/s 2 )
If the final velocity is smaller than the initial velocity, the object must be slowing down. An object that is slowing down has a negative acceleration and we show this by putting a negative sign in front of the acceleration.
Example 5: Grace is driving her sports car at 30 m/s when a ball rolls in front of her. Grace slams on the brakes and comes to a stop in 3.0 s.
What was the acceleration of Grace’s car?
Solution:
Given: v
v i f
= 30 m/s
= 0 m/s
Δt = 3.0 s Δt
Solve: a = v f
– v i
= 0 m/s – 30 m/s = -10 m/s 2
Δt 3.0 s unknown: a = ? equation: a = v f
– v i
The negative sign means that she was slowing down.

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Practice Exercises:
Please follow the format written in your notebook.
1.
An airplane taxiing down the runway receives word that it must return to the gate to pick up an important passenger who was late to his connecting flight. The jet is travelling at 45.0 m/s when the pilot receives the message. What is the acceleration of the plane if it takes the pilot 5.00 s to bring the plane to a stop?
2.
While driving her supercar at 20.0 m/s down the Canyon E-way,
Shelby comes up behind a slow moving school bus and decides to pass it. If Shelby can accelerate at 5.00 m/s 2 , how long will it take her to reach a speed of 30.0 m/s?
3.
Vivian is walking to the hairdresser’s at 1.3 m/s when she glances at her watch and realizes that she is going to be late for her appointment. Vivian gradually quickens her pace at a rate of
0.090 m/s 2 . a.
What is Vivian’s speed after 10.0 s?
Free Fall b.
Convert this speed to mi/hr (1600 m = 1 mi) c.
Is she walking, jogging, or running very fast?
Free fall: The movement of an object in response to a gravitational attraction.
When an object is released, it falls towards the earth due to gravitational attraction the earth provides. As the object falls, it will accelerate at a constant rate of -9.8 m/s 2 regardless of its mass.
There are many different ways to solve free fall problems. The sign convention used may be different depending on who is teaching. In this class we will use a negative sign to show that the acceleration is going downward. Be aware that if something is starting out going upwards, that speed is positive, whereas if it starts out going downwards the velocity will be negative.

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* Down, left, or coming back is negative
* Up, right, or leaving is positive
The displacement of falling object in a given amount of time is written as:
Δd = v i
Δt + gΔt 2
2
The final velocity of a falling object can be represented by the equation: v f
2 = v i
2 + 2gΔd or by the earlier equation, a = v f
– v i
rewritten as v f
= v i
+ aΔt or:
Δt v f
= v i
+ gΔt
Note that the term “g” in all of these exercises can be written as “a” if you use a constant acceleration other than gravity. Therefore, these equations can be used for objects moving in any direction.
We neglect air resistance in most free fall exercises (including those in this book), although in real life, air resistance is a factor that must be taken into account. This book will also assume that the initial speed of all objects in free fall is zero, unless otherwise specified.

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Example 1: King Kong carries Fay Wray up the 321 m tall Empire state building. At the top of the skyscraper, Fay Wray’s shoe falls from her foot.
How fast will the shoe be moving when it hits the ground?
Solution:
Given: v i
= 0 m/s
g = -9.8 m/s 2
Δd = 321 m unknown: v equation: v f f
2
= ?
= v i
2 + 2gΔd
Solve: v f
= √v i
2 + 2gΔd = √0 + 2(-9.8 m/s 2 )(321 m) = √-6291 m 2 /s 2 you can’t square root a negative, so bring it to the outside of the square root.
= -√6291 m 2 /s 2 = -79.3 m/s
Example 7: The Steamboat Geyser in Yellowstone National Park, WY is capable of shooting its hot water up the ground with a speed of 48.0 m/s.
How high can this geyser shoot?
Solution: Remember, the geyser is shooting up, so it will have a positive initial velocity, while gravity is still negative. At the top of the geyser the velocity is zero, right before it turns around and falls down, so the final velocity is zero in this case. given: v i
= 48 m/s
v f
= 0 m/s
g = -9.8 m/s 2 unknown: Δd = ? equation: v f
2 = v i
2 + 2gΔd
Solve: Δd = v f
2 – v i
2 = (0 m/s) 2 – (48.0 m/s) 2 = 117.6 m
2g 2(-9.8 m/s 2 )

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Example 8: A baby bird sits in a tree waiting for dinner.
As its mother arrives back to the nest, she drops a worm towards the hungry chicks mouth. She misses and the worm falls to the ground in 1.5 s. How high up is the nest?
Solution: When something is dropped, we assume the initial velocity is zero.
Given: v i
= 0m/s
g = -9.8 m/s2
t = 1.50 s unknown: Δd = ? equation: Δd = v i
Δt + gΔt 2
2
Solve: Δd = v i
Δt + gΔt 2 = 0 + (-9.8 m/s 2 )(1.5 s) 2 = -11.0 m
2 2
The distance is negative because it’s falling down.
Example 9: A giraffe, who stands 6 m tall, bites a branch off a tree to chew on the leaves, and he lets the branch fall to the ground. How long does it take the branch to hit the ground?
Solution: given: Δd = -6 m
g = -9.8 m/s2
vi = 0 m/s unknown: Δt = ? equation: Δd = viΔt + gΔt2
2 solve: Δt = 2Δd = 2(-6 m) = √1.2 s 2 = 1.11 s
√ g √ -9.8 m/s 2

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Practice Exercises:
Please follow the format given on the board.
1.
Billy, a mountain goat, is rock climbing on his favorite slope one sunny spring morning when a rock comes falling toward him from a ledge 40.0 m above. Billy ducks and avoids injury. a.
How fast is the rock traveling when it passes Billy? b.
How does this speed compare to a car traveling 25 m/s (the equivalent of 55 mi/hr)?
2.
Pastor Rick climbs to the church belfry one morning to determine the height of the church. He drops a book and observes that it takes
2.00 s to strike the ground below. a.
How high is the church belfry? b.
Why should the pastor drop a whole book instead of just one page?
3.
How long is Tina the ballerina in the air if she leaps straight up into the air with an intial speed of 1.8 m/s?
f
4.
A seagull drops a clam from a height of 25 m, how long will it take the clam to fall?
5.
At Six Flags Over Texas, there’s a free fall ride that carries passengers up to a height of 33.5 m and drops them to the ground.
How fast are the passengers going at the bottom of this exciting ride?
6.
Aliens are playing basketball on the planet Zarth. During the game a player flies above the basket and drops the ball in from a height of 10 m. If it takes the ball 5.0 s to fall, what is the acceleration due to gravity on planet
Zarth?
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