Free-Fall Motion
“Free-fall” motion is a Uniformly Accelerated Motion that takes place in a vertical direction. Anytime an object moves vertically, either going upwards, or going downwards, we say it is in Free-fall. There are two key, critical concepts that we must understand when discussing objects in Free-fall motion:
1. The constant acceleration of the object is ALWAYS directed downwards due to the influence of Earth’s gravity; the acceleration due to gravity on Earth is referred to as “ g ”, and has a value of 9.8 m/s 2 . Because of sign conventions, where we choose up to be the positive direction and down to be the negative, we will say that, in Free-fall, the vertical acceleration of an object is -g = -9.8 m/s 2 .
[ Recall: the negative sign indicates that the direction of the acceleration is downwards.
]
2. When an object is in Free-fall, we will always ignore any effects that air resistance may have on the object’s motion. To demonstrate, take a sheet of paper, and allow it to fall to the floor. Due to air resistance, the paper will twist and turn, gliding back and forth, slowly downwards to the ground. Now, drop a pen to the floor. Notice that it falls quickly, straight down to the ground.
We will assume that objects fall more like pens than pieces of paper. [ For a demonstration of a feather and steel ball being dropped in air, and then without air resistance, view this YouTube video: ] http://www.youtube.com/watch?v=_XJcZ-KoL9o
Again, understand that the acceleration will ALWAYS be directed downwards. For instance, when a tennis ball is tossed upwards into the air, its initial velocity is directed upwards, and would be considered to be positive. As soon as it leaves your hand, it will begin to accelerate downwards
( implying that its velocity will decrease; this means it will start to slow down ), at a rate of -9.8 m/s 2 .
Eventually, the ball will stop moving; at the instant it has stopped moving, its velocity is zero, and it has reached its greatest vertical position ( simply stated, it will not go higher than this position; we call this is Maximum Height ). However, it is still accelerating downwards, at a rate of -9.8 m/s 2 ! ( If it didn’t accelerate, and it had zero velocity, would it ever fall back down?
) So, it begins to fall back down; this implies its velocity will now be negative, because down is considered the negative direction. Because the acceleration is also negative, this tells us that the velocity is decreasing; perhaps confusingly, we will find in this case that the tennis ball is going to speed up... although the velocity is still negative ( you could say that its velocity is becoming more negative as it continues to accelerate downwards ).
Notation: In order to reinforce that Free-fall is a vertical motion ( as opposed to the Horizontal motions we’ve discussed so far ), we make a few small notational changes:
● x for position will be replaced by y ( because the vertical axis is the y-axis )
● Δx for displacement will be replaced by Δy
● The subscript “y” will be added to other variables; v
0y
, v fy
, a y
( note that a y
= -g = -9.8 m/s 2 )
Facts: The following are always true for Free-fall motion...
● The acceleration is -9.8 m/s 2 regardless of the direction of the object’s motion
● The velocity is positive when the object is moving upwards, and negative when the object is moving downwards
● If an object is “dropped”, its initial velocity v
0y
= 0
○ implying its initial position is the highest point it will reach, and its displacement and final velocity must be negative

● The velocity is zero when the object reaches its highest point
● When an object is tossed upwards with a velocity +v
0y
and returns back down to its starting point,
○ its final velocity v fy
will be equal to -(v
0y
)
This implies that the speed going up matches the speed coming down; the opposite sign indicates the change in direction
○ the time it takes to get from the bottom up to the highest point ( this is sometimes called t up
) is equal to the time it takes to get from the highest point back down to the bottom
( this is sometimes called t down
)
Equations for Free-Fall Motion: Note that the format of the equations is the same as kinematics; the only difference is a small adjustment with notation. Recall: a y
= -9.8 m/s 2 !

Examples:
1. A tennis ball is tossed upwards into the air with an initial velocity of +5 m/s.
a. What is the highest point the tennis ball will reach?
b. How much time does it take for the tennis ball to reach this point?
The vertical displacement from where the tennis ball starts is ~1.28 m; this indicates that the final position is 1.28 m above the initial position. If I assume the initial position is zero, then my final position would equal the displacement: 1.28 m above the starting height.

2. A rock is dropped, and falls for a time of 1.0 seconds.
a. What is its velocity at this time?
b. How far has it fallen?
This should not be a surprise; the acceleration implies that the velocity should decrease by 9.8 m/s for each second that it moves. It moves for 1.0 s, so its velocity should go down by 9.8 m/s, and we started at zero.
This implies that, in just 1.0 seconds, an object dropped from rest will fall 4.9 meters (that is more than
15 feet!).