Notes on Motion VI

Free FallA Special type of uniform acceleration

Free FallWhen an object is only allowed to move under the influence

of gravity.

When you drop a ball, it is in free fall!

When you throw a ball downward, it is in free fall! (After it is released)

When you throw a ball upward, it is in free fall! (After it is released)

Free FallWhen an object is in free fall, it has a uniform acceleration.

Objects in free fall are ALWAYS accelerated downward by gravity.

On Earth, objects in free fall are accelerated downward at a rate of 9.8 m/s/s (or 9.8 m/s2).

When we are working with formulas we use “g” to represent the acceleration due to gravity.

g = 9.8 m/s2

In the English System of measurement g = 32 ft/s2.

Free Fall Speed/VelocityIn general, the formula we use to find the velocity of an object in

free fall is:

atvv if The acceleration of an object in free fall is the acceleration due to

gravity, g, and is downward so a = -g = -9.8-m/s/s.

9.8tvv if

Rewriting to find time, you get:

9.8

vvt if

The speed of an object in free fall is just the velocity without the sign.

9.8tvv if

9.8

vvt if

Example Problem 1A cliff diver dives from the top of a cliff. If he is able to fall for 10-s, what will his speed be when he hits the water?

Useful Formulas

Know Find

Formula Solution

vi = 0-m/st = 10-sFree Fall

vf = ?

sm

f

f

98v

9.8(10)0v

Since we are only asked for speed, we can ignore the (-), so the speed is 98-m/s.

9.8tvv if

9.8

vvt if

Example Problem 2A ball is thrown upward into the air at 35-m/s. How long will it take the ball to reach its maximum height?

Useful Formulas

Know Find

Formula Solution

vf = 0-m/svi = 35-sFree Fall

t = ?

st

t

6.38.9

35

8.9

350

What would the hang time of this ball be?Since it takes 3.6-s to reach its maximum height, it will take an

additional 3.6-s to reach the ground again.3.6 + 3.6 = 7.2-s

Distance in Free FallWhen an object that started from rest had a uniform acceleration,

we had an equation for the distance it travelled in a given time:

221 atd

But since we know the acceleration of the object is “g” (9.8-m/s2), we can change this to:

2

221

t9.4d

t)8.9(d

If we need to find the time it takes an object to fall a certain distance, we can solve this equation for t:

9.4

dt

9.8tvv if

9.8

vvt if

Example Problem 3A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How far will this rock have fallen in 5-s?

Useful Formulas Know Find

Formula Solution

t = 5-sFree Fall

d = ?

m5.122d

)25(9.4d

)5(9.4d 2

2t9.4d

9.4

dt

How far from the ground will the rock be at this time?

h = 381 - 122.5 = 258.5-m

9.8tvv if

9.8

vvt if

Example Problem 4A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How long will it take for the rock to hit the ground?

Useful Formulas Know Find

Formula Solution

d = 381-mFree Fall

t = ?

s8.8t

76.77t

9.4

381t

2t9.4d

9.4

dt

9.8tvv if

9.8

vvt if

Example Problem 5A rock is dropped from the top of the Empire State Building in New York City. The height of this building is 381-m. How fast will the rock be travelling just as it hits the ground?

Useful Formulas Know Find

Formula Solution

t = 8.8-sFrom Example 4

Free Fall

vf = ?

sm

f

f

2.86v

)8.8(8.90v

2t9.4d

9.4

dt Since we are only

asked for speed, we can ignore the (-), so the speed is 86.2-m/s.

FYI: 86.2-m/s is approximately 194-mph. That could do some damage.

Finding Maximum HeightWhen an object is thrown upward, we know that the time it takes for it to reach its maximum height is the same as the time for it to

fall back to the ground from that height.

We can use this information to help us find the maximum height of the object.

FirstFind the time it takes to reach its

maximum height using the formula:

9.8

vvt if

SecondUse the time found in the first step to calculate the distance it falls from its

maximum height. Use:

2t9.4d

We also know that its velocity at its maximum height is 0.

9.8tvv if

9.8

vvt if

Example Problem 6A rifle fires a bullet at 896-m/s. If the bullet is fired straight up, find its maximum height.

Useful Formulas Know Find

Formula Solution

vi = 896-m/svf = 0-m/sFree Fall

t = ?

s91t8.9

8960t

2t9.4d

9.4

dt

First find the time to maximum height:

NOWUSE THIS TIME TO FIND THE MAXIMUM HEIGHT

9.8tvv if

9.8

vvt if

Example Problem 6 ContinuedA rifle fires a bullet at 896-m/s. If the bullet is fired straight up, find its maximum height.

Useful Formulas

Know Find

Formula Solution

t = 91-sFree Fall

d = ?

m577,40d

)8281(9.4d

)91(9.4d 2

2t9.4d

9.4

dt