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Faith is to believe what you do not see; the reward of this faith is to see what you believe.
St. Augustine
A car will accelerate from a red light. A runner will accelerate after hearing the starter’s gun. A ball will accelerate after it is dropped. The acceleration of the ball will occur due to the force of gravity. A force is something that pushes or pulls on an object. In this case, it is that force that pulls the ball towards the ground.
The diagram to the left shows the position of a ball that has been dropped from a ledge at 0.1 s intervals during its fall. You can see that during each successive 0.1 s interval, the ball has moved a greater distance than it had in the previous interval. This tells us that the speed of the ball is increasing – it is undergoing an acceleration.
Hundreds of years ago, people believed that heavy objects fell more quickly than light objects due to the fact that they were heavier.
Famous People
According to legend, Galileo refuted this claim and proved it to be false by dropping objects of different masses over the side of the Leaning Tower of Pisa and observing that they fell at the same rate. He also presented the following argument to support his claim:Imagine two identical 1 kg bricks being dropped at the same time. They have the same mass and are identical in every way, so it would be expected that they would fall at the same rate.Now, what would happen if we placed the two identical 1 kg brick beside each other, almost touching, and repeated the experiment? Again, it would be expected that they would fall at the same rate.Now, what would happen if we tied a very narrow piece of string around the bricks and repeated the experiment? They would still fall at the same rate as they had before. The fact that they now formed a larger, heavier 2 kg object would not cause them to fall any faster.
Prior to Galileo, people believed that heavier objects would fall more quickly than lighter ones because this is what they observed in everyday life. Hammers fell to the ground faster than feathers. Though this may be true, it is not because of the fact that hammers are heavier. It occurs because of the amount of air resistance acting on the object. If we could remove the air resistance, all objects would fall at the same rate.
Watch this video to see if Galileo was correct about the hammer vs. the feather!
It is very difficult to account for air resistance in motion calculations because the amount of air resistance acting will change as the speed of the object changes. Generally the solution to such problems requires the use of calculus. To avoid this difficulty, unless otherwise stated, you will ignore the effects of air resistance for any calculus in this course. We can do this because the effects of air resistance are minimal until we get to high speeds.
You have seen that, if we ignore air resistance, all objects fall at the same rate. The next step is to determine this rate of acceleration
On Earth the acceleration of an object in free-fall due to gravity is 9.8 m/s2 [down]. We can use this value of acceleration in any of the motion equations to solve problems.
Example
A diver steps off the edge of a 10 m diving platform and starts to fall towards the water below.How long will it take the diver to hit the water?How fast will the diver be moving when she hits the water?
For most gravity problems, it is best to make all upward motion positive and all downward motion negative.
Given:
. This is negative because the person travels 10 m down.
Required:
Solution:
, since , the term 
is eliminated from the equation.
It will take the diver 1.43 s to hit the water and she will be moving at a velocity of 14.0 m/s [down]. This is about 50 km/h.
Practice Questions
- Jennifer throws a baseball straight up into the air with a velocity of 16 m/s from the balcony of her apartment 20 m above the ground.a. What is the maximum height of the baseball?b. At this height, what is the velocity of the baseball?c. At this height, what is the acceleration of the baseball?d. For how long will the baseball be in the air before hitting the ground below?e. With what velocity will the baseball hit the ground?
Answer
In this solution, all motion up is positive; all motion down is negative
- What is the maximum height of the baseball?
Given:
As shown in the diagram, the ball reaches its maximum height when the velocity of the ball reaches 0 m/s.
. This must be true because at the maximum height, the baseball will no longer be moving upwards.
Required:
This will give the height of the baseball above the balcony. This value must then be added to the 20 m initial height of the balcony.
We will need to find this value before we can solve for maximum height.
Solution:
The maximum height of the baseball will be 33 m. (13 m + 20 m = 33 m)
- At this height, what is the velocity of the baseball?
At this height, the velocity of the baseball will be 0 m/s. The baseball cannot be moving upwards or it will not yet have attained its maximum height. The baseball cannot be moving downwards or it will have already passed its maximum height.
- At this height, what is the acceleration of the baseball?
The acceleration of the baseball is -9.8 m/s2. This is the acceleration of the baseball from the moment it leaves Jennifer’s hand until the moment it touches the ground.
- For how long will the baseball be in the air before hitting the ground below?
From “part a” we know that it takes 1.63 s for the baseball to reach its maximum height. Now all you need to do is find how long it takes the ball to go from maximum height to the ground.
Given:
Required:
?t = ?
Solution:
, since , the term is eliminated from the equation.
?t = 2.60 s
The baseball will be in the air for 4.23 s before hitting the ground; the time taken for the upward motion plus the time taken for the downward motion (1.63 s + 2.60 s= 4.23 s).
- With what velocity will the baseball hit the ground?
Given: (Be very careful with signs here.)
?t = 4.23 s
Required:
Solution:
When the baseball hits the ground, it will be moving down at 25.5 m/s.