KAHIBARO
Table of Contents
Gravity as the Cause of Falling Motion
Free fall is the motion of an object when gravity is the only force acting on it. Near the surface of Earth, this means the object accelerates downward because of Earth's gravitational pull. In an ideal free fall description, we ignore air resistance. A dropped stone, a ball thrown upward, and an object tossed downward can all be treated as free fall as long as only gravity is considered.
The key idea is that the acceleration is constant and points downward. Near Earth's surface, this acceleration has magnitude
$$
g \approx 9.8\ \text{m/s}^2
$$
This quantity is called the acceleration due to gravity.
In ideal free fall near Earth's surface, the acceleration is constant and downward:
$$
a = -g
$$
if upward is chosen as the positive direction.
Choosing a Sign Convention
To solve free fall problems, it is very important to choose a coordinate direction. A common choice is to take upward as positive. Then gravity acts downward, so acceleration is negative:
$$
a = -g
$$
If instead downward is chosen as positive, then
$$
a = +g
$$
Both choices are correct, but you must stay consistent throughout the calculation.
Free Fall from Rest
If an object is simply dropped, its initial velocity is zero:
$$
v_0 = 0
$$
With upward positive, the kinematic equations for free fall become
$$
v = v_0 - gt
$$
$$
y = y_0 + v_0 t - \frac{1}{2}gt^2
$$
$$
v^2 = v_0^2 - 2g(y - y_0)
$$
For a dropped object, since $v_0 = 0$, these reduce to
$$
v = -gt
$$
$$
y = y_0 - \frac{1}{2}gt^2
$$
This shows that the object moves farther and farther downward as time passes, and its speed increases linearly with time.
For an object dropped from rest, with upward positive:
$$
v = -gt
$$
$$
y = y_0 - \frac{1}{2}gt^2
$$
Throwing an Object Upward
Free fall also includes motion upward after an object is thrown. Even while the object rises, gravity still acts downward, so the acceleration remains the same. The upward velocity gradually decreases until it becomes zero at the highest point.
If the initial velocity is upward, then $v_0 > 0$, and
$$
v = v_0 - gt
$$
At the top of the motion,
$$
v = 0
$$
so the time to reach the highest point is
$$
t_{\text{top}} = \frac{v_0}{g}
$$
This does not mean gravity disappears at the top. Only the velocity becomes zero for an instant. The acceleration is still downward.
At the highest point of vertical free fall, the velocity is zero for an instant, but the acceleration is still:
$$
a = -g
$$
Throwing an Object Downward
If an object is thrown downward, it already has an initial downward velocity. With upward chosen as positive, that means $v_0$ is negative. Gravity then increases the downward speed further.
The same equations still apply:
$$
v = v_0 - gt
$$
$$
y = y_0 + v_0 t - \frac{1}{2}gt^2
$$
The form of the equations does not change. Only the signs of the quantities change according to the chosen direction.
Physical Meaning of $g$
The value $g \approx 9.8\ \text{m/s}^2$ means that the velocity changes by $9.8\ \text{m/s}$ every second. If an object falls downward from rest, after 1 second its velocity is about $9.8\ \text{m/s}$ downward, after 2 seconds about $19.6\ \text{m/s}$ downward, and so on, as long as air resistance is neglected.
The value of $g$ is approximately constant only near Earth's surface. In more advanced situations, gravity can vary with altitude, but that is not needed for basic free fall.
Free Fall and Mass
In ideal free fall, all objects have the same acceleration regardless of mass. A heavy object and a light object fall with the same acceleration if air resistance is ignored.
This is why, in vacuum, different objects dropped from the same height reach the ground at the same time.
| Object | Mass | Ideal free fall acceleration |
|---|---|---|
| Tennis ball | small | $g$ downward |
| Stone | larger | $g$ downward |
| Hammer | much larger | $g$ downward |
The mass changes the weight force, but not the free fall acceleration in this ideal model.
Typical Free Fall Situations
Many basic problems can be recognized as free fall situations.
| Situation | Initial velocity | Acceleration |
|---|---|---|
| Object dropped | $v_0 = 0$ | downward, magnitude $g$ |
| Object thrown upward | $v_0 > 0$ if upward is positive | downward, magnitude $g$ |
| Object thrown downward | $v_0 < 0$ if upward is positive | downward, magnitude $g$ |
The common feature in all cases is that gravity is the only force considered.
Visualizing Free Fall
The velocity changes steadily while the acceleration stays constant.
This drawing shows that the acceleration points downward at every stage of the motion.
A Simple Example
Suppose a ball is dropped from rest. After $2.0\ \text{s}$, its velocity is
$$
v = -gt = -(9.8)(2.0) = -19.6\ \text{m/s}
$$
The negative sign means the velocity is downward, if upward is positive.
Its change in position during that time is
$$
y - y_0 = -\frac{1}{2}gt^2
$$
$$
y - y_0 = -\frac{1}{2}(9.8)(2.0)^2 = -19.6\ \text{m}
$$
So the ball is $19.6\ \text{m}$ below its starting point after 2.0 seconds.
Common Mistakes
A very common mistake is to think that acceleration becomes zero when an object reaches the highest point. This is false. The velocity becomes zero there, but gravity still acts, so acceleration remains downward.
Another common mistake is to ignore signs. If upward is positive, then downward velocity and downward displacement are negative.
Do not confuse velocity and acceleration.
At the top of the motion:
$$
v = 0
$$
but
$$
a = -g
$$
Free Fall on Other Worlds
The idea of free fall is the same everywhere, but the value of gravitational acceleration can change from one planet or moon to another.
| World | Approximate gravitational acceleration |
|---|---|
| Earth | $9.8\ \text{m/s}^2$ |
| Moon | $1.6\ \text{m/s}^2$ |
| Mars | $3.7\ \text{m/s}^2$ |
So an object falls more slowly on the Moon than on Earth, because the downward acceleration is smaller.
Final Idea
Free fall is one of the simplest and most important examples of motion with constant acceleration. Once gravity is treated as a constant downward acceleration, the motion can be described with the same kinematic equations used for any constant acceleration problem, with $a$ replaced by $\pm g$ depending on the chosen axis.