2.7 Falling Objects

Gravity and Falling Objects

Near Earth's surface, all objects in free fall accelerate downward at the same rate, regardless of mass. This constant acceleration due to gravity is the starting point for analyzing any problem where something is dropped, thrown up, or launched vertically.

Effects of Gravity on Falling Objects

Gravity is a force that attracts objects with mass toward each other. On Earth, this means everything gets pulled toward the planet's center. The force of gravity acting on an object's mass is what we call weight.

The key quantity here is the acceleration due to gravity, written as $g$:

$g = 9.8 \, m/s^2 \quad (32 \, ft/s^2)$

This value is constant near Earth's surface and does not depend on the object's mass. In a vacuum (where there's no air resistance), a feather and a bowling ball fall at exactly the same rate. In everyday life, air resistance makes lighter objects fall more slowly, but the underlying gravitational acceleration is the same for both.

Motion Analysis in Free Fall

An object in free fall starts with some initial velocity ($v_0$) and accelerates downward at $g$. If you simply drop an object, $v_0 = 0$.

Velocity at any time $t$:

$v = v_0 + gt$

If dropped from rest, this simplifies to $v = gt$. For example, a ball dropped from a tower has a velocity of $9.8 \, m/s$ after 1 second, $19.6 \, m/s$ after 2 seconds, and so on. Velocity increases linearly with time, so a graph of $v$ vs. $t$ is a straight line.

Displacement at any time $t$:

$\Delta y = v_0 t + \frac{1}{2}gt^2$

If dropped from rest, this simplifies to $\Delta y = \frac{1}{2}gt^2$. A rock dropped from a cliff falls about $4.9 \, m$ in the first second and $19.6 \, m$ in the first two seconds. Displacement increases quadratically, so a graph of $\Delta y$ vs. $t$ is a parabola.

Remember that velocity is a vector: it has both magnitude and direction. A positive velocity might mean upward, and a negative velocity means downward (or vice versa, depending on your chosen sign convention). Stay consistent.

Energy Considerations in Free Fall

As an object falls, it loses gravitational potential energy and gains kinetic energy. The total mechanical energy (potential + kinetic) stays constant as long as air resistance is negligible. This is a preview of energy conservation, which you'll study in more depth later. For now, just recognize that falling converts height into speed.

Problem-Solving for Free-Falling Objects

Three kinematic equations cover every free-fall problem:

1. $v = v_0 + gt$
2. $\Delta y = v_0 t + \frac{1}{2}gt^2$
3. $v^2 = v_0^2 + 2g\Delta y$

Each equation connects a different set of variables. Pick the one that includes your unknown and the quantities you already know.

Steps for solving a free-fall problem:

1. Draw a quick sketch and choose a positive direction (up or down).
2. List what you know: $v_0$, $v$, $t$, $\Delta y$, and $g = 9.8 \, m/s^2$.
3. Identify the unknown you need to find.
4. Select the kinematic equation that contains your unknown and your known quantities.
5. Substitute values and solve. Use consistent units (meters and seconds).
6. Check the sign of your answer. If you chose "up" as positive, a negative displacement means the object moved downward.

Example: A ball is thrown straight up with an initial velocity of $20 \, m/s$. How long does it take to reach its maximum height?

At maximum height, the ball's velocity is momentarily zero, so $v = 0$. Gravity acts downward, so use $g = -9.8 \, m/s^2$ (taking up as positive).

Using equation 1:

$0 = 20 + (-9.8)t$

$t = \frac{20}{9.8} \approx 2.04 \, s$

The ball takes about 2 seconds to reach its peak. Notice that $g$ is negative here because gravity opposes the upward motion. Getting the sign of $g$ right is one of the most common places students make mistakes, so always define your positive direction first.