Physics 11 Date: ______
Lesson 6: Acceleration Due To Gravity
When objects fall towards the Earth (or any other large massive object), they experience an acceleration downwards towards the surface of the planet. Think about it … if you drop your pen, where will it go? Straight downwards unless acted upon by a force!! If the ground did not stop the object, it would actually carry on to the center of the Earth.
Galileo, in the 1600s, was the first to state that all objects fall towards the Earth at a constant acceleration. For example, he said, a kilogram of lead will fall at the same acceleration rate as a single piece of paper or a feather. During this time, most people had a very hard time believing this idea. Everyone assumed that heavier objects would fall faster. Galileo proved his ideas through experiments and demonstrations, reportedly by dropping objects from the Leaning Tower of Pisa. Galileo had to be very careful about his ideas since they challenged those held by prevailing rulers or church leaders. He met with threats of imprisonment and was actually imprisoned for teaching that the Earth was not the center of the universe.
The academic community later accepted Galileo’s notion about the acceleration due to gravity. As a final proof of his idea, American astronauts in the early 1970s actually demonstrated that a feather and a hammer fall exactly at the same rate on the moon … hitting the lunar ground at the same time (in a situation when air resistance cannot slow down the movement of the feather since there is no atmosphere on the moon).
We now know the value of the acceleration due to gravity as 9.80 m/s2 near the surface of the planet (see Physics 12 for Newton’s Universal Gravitational equation for objects at extreme distances!). We sometimes use the value as a = 9.80 m/s2 and sometimes even give it a special letter g, therefore g = 9.80 m/s2. When Physics students analyze problems involving falling objects, the same formulas are used as in all kinematics problems:
These formulas work in exactly the same way when objects fall downwards or upwards as when they are moved horizontally. However, there is one subtle but large difference to remember. We must assign either a positive or negative sign to the acceleration depending upon how we set up the frame of reference. For example, if we want to know how high an object will climb upwards after being shot out of a cannon at 50.0 m/s, we must think about the directions of the velocity and the acceleration. If the acceleration is in an opposite direction to the velocity, one of these variables must be assigned a negative value. Usually, objects moving upwards are given a positive velocity while the downward acceleration is negative. We often say that things moving to the right are positive, while we consider movement to the left as negative. The same thing is true when we consider up and down, positive for up while negative for down. It is relatively easy after some practice.
One other thing to think about … consider an object thrown directly upwards. As you throw the object upwards at some initial velocity, it begins to slow down (because it is accelerating back towards earth by gravity), it then stops for a brief instant (velocity equals zero) when it reaches its maximum height and then begins to fall downward accelerating constantly and increasing its speed. Therefore, we need to separate out the motion of the object into several parts and examine each separately e.g. motion upwards until it reaches maximum height and then, the second part, the motion as it falls from that maximum height towards the ground. In this course, free falling objects are objects that do not encounter air resistance.
Here’s the rest of the example to illustrate:
A ball is shot directly upwards at 50.0 m/s out of a cannon. To what height will this ball reach AND how long will it take to reach this height.
For the distance use
Note that vf will be zero at the maximum height!
If you know that vi is 50.0 m/s and that a = -9.80 m/s2, you can easily determine the distance the object goes upwards (the maximum height)
For the time to reach the maximum height use:
Note that again vf will be zero at the maximum height.
If you know that vi is 50.0 m/s and that a=-9.80 m/s2, you can easily determine the time the object goes upwards to the maximum height:
Note that if you were asked to find the total time in the air, you would simply double this time you just calculated.
The position vs. time graph for a free-falling object is shown below.
The velocity vs. time graph.
When determining how fast an object is traveling after it is dropped from rest, we use v = gt. To determine how far it has traveled we use d = ½ gt2 since vi = 0.
Here are some key points for free falling questions:
- If an object is dropped (as opposed to being thrown) from an elevated height to the
ground below, the initial velocity of the object is 0.
- If an object is projected upwards in a vertical direction, it will slow down as it rises
upward. The instant at which it reaches the peak of its trajectory, its velocity is 0.
- If an object is projected upwards in a vertical direction, then the velocity at which it is
projected is equal in magnitude and opposite in sign to the velocity it has when it
returns to the same height.
Examples:
1. A rubber ball is dropped from rest. Determine it’s velocity at 1, 2, 3, and 4 seconds.
2. A ball is thrown upward at an initial velocity of 29.40 m/s. Determine it’s velocity at 1,
2, 3, 4, 5 and 6 seconds.
3. A baseball thrown vertically upward from the roof of a tall building has an initial
velocity of 20 m/s.
a) Calculate the time required to reach its maximum height.
b) Determine the maximum height.
c) Determine its position and velocity at 1.5 s.
d) Determine its position and velocity after 5s.
5. Bea O’Problem drops a roof shingle from the top of a roof located 8.52 m above the
ground. Determine the time required the a shingle to reach the ground.
6. Rex Things throws his mother's crystal vase vertically upwards with an initial velocity of
26.2 m/s. Determine the height to which the vase will rise above its initial height.
7. A team of students competing in the "Egg Drop" competition at the UBC physics
Olympics at UBC, drop their egg from a ledge, three floors above the ground.
a) If it takes the egg 2.4 seconds to reach the ground, calculate the height
of the ledge in metres.
b) How fast was the egg traveling the instant before it hit the ground?
Additional Problems:
1. A ball is dropped from a tower 70.0 m high. How far will the ball have fallen after 3
seconds?
2. A ball is thrown downward with an initial velocity of 3.00 m/s. What would be its position after 2.00 s? What would its speed be after 3.00 s?
3. A person throws a ball upward into the air with an initial velocity of 15.0 m/s. Calculate how high it goes. Calculate how long the ball is in the air before it comes back to his hand.
4. If a ball is thrown upwards with an initial velocity of 10.09 m/s, determine its maximum height.
5. Brett is riding the Giant Drop at Great America. If Brett free-falls for 2.6 seconds,
what will be his final velocity and how far will he fall?
6. A kangaroo is capable of jumping to a height of 2.62 m. Determine the take-off speed of the
Kangaroo.
7. A baseball is popped straight up into the air and has a hang-time of 6.25 s. Determine the
height to which the ball rises before it reaches its peak. (Hint: the time to rise to the peak is
one-half the total hang-time.)
8. The observation deck of a skyscraper is 420 m above the street. Determine the time required
for a penny to free-fall from the deck to the street below.
9. A goat falls over a cliff. The height of the cliff is 140 m. Determine the goats’ velocity the moment before it hits the water below.
10. Shelby pushes Trent over a cliff. Trent has 12 s to say his prayers. How high is the cliff?
11. Stephen shoots an arrow up in the air. The arrows velocity right after its release is 200 m/s.
Determine the maximum height of the arrow? Determine how long it is in the air?
12. An object is dropped from rest on the surface of the Moon. The acceleration due to gravity
on the Mon is 1.6 m/s2. What is the speed of the object after it has dropped 54 m?
13. Ricky Steamboat jumps from a 10 m high bridge into the swirling waters below.
a) What is his speed as he hits the water?
b) How long does it take Ricky to hit the water?
14. Matt shoots an emergency flare straight up into the air with an initial velocity of 60 m/s.
(assume air resistance is negligible).
a) How long does it take to achieve maximum height?
b) Determine the flare’s maximum height.
c) How long does it take for the flare to return to earth?
15. If Michael Jordan has a vertical leap of 1.29 m, what is his take-off speed and his hang time
(total time to move upwards to the peak and then return to the ground)?
16. Ivana Tinkle throws a ball up in the air. Determine how long it takes for the ball to reach a maximum height of 8.0 m above her hand.