Q.Free-fall: Discuss the motion of an object under free fall. Neglect air resistance.

Concept: Projectile Motion Under Gravity (Free Fall as 1-D Special Case)

In free fall, the only force acting is gravity, giving a constant downward acceleration $g \approx 9.8 \, \text{m/s}^2$. Since air resistance is neglected, the motion is purely vertical.

1. Acceleration: Constant, $a = -g$ (taking upward as positive).
2. Velocity: $v = u - gt$, where $u$ is initial velocity.
3. Displacement: $s = ut - \frac{1}{2}gt^2$. …

Concept and Intuition

Free fall is a special case of projectile motion where the only force acting on an object is gravity. When we neglect air resistance, the object experiences a constant downward acceleration of magnitude $g$ (approximately $9.8 \, \text{m/s}^2$ near Earth's surface). This is a beautiful simplification: the motion becomes uniformly accelerated motion in the vertical direction.

The most famous demonstration of this principle is Galileo's Leaning Tower of Pisa experiment (whether apocryphal or not) — a heavy ball and a light ball dropped from the same height hit the ground at the same time. This is because gravitational acceleration is independent of mass. In a vacuum, a feather and a hammer fall identically, as shown on the Moon during the Apollo 15 mission.

For free fall, we typically take the downward direction as positive (or upward as positive — be consistent). The equations of motion become:

• $v = u + gt$ (velocity after time $t$)
• $s = ut + \frac{1}{2}gt^2$ (displacement after time $t$)
• $v^2 = u^2 + 2gs$ (relation between velocity and displacement)

where $u$ is initial velocity, $v$ is final velocity, $s$ is displacement, and $t$ is time.

Step-by-Step Analysis

1. Define the sign convention and parameters.

Let's take the downward direction as positive. This means:

• Acceleration $a = +g = 9.8 \, \text{m/s}^2$
• If the object is dropped from rest, initial velocity $u = 0$
• If thrown downward, $u$ is positive
• If thrown upward, $u$ is negative (since upward is opposite to our positive direction)

2. Consider the case of an object dropped from rest.

This is the simplest free-fall scenario. The object starts with zero initial velocity and falls straight down.

Using $s = ut + \frac{1}{2}gt^2$ with $u = 0$:

The distance fallen is proportional to the square of the time. After 1 second, the object falls about $4.9 \, \text{m}$; after 2 seconds, about $19.6 \, \text{m}$; after 3 seconds, about $44.1 \, \text{m}$.

The velocity after time $t$ is:

So after 1 second, $v = 9.8 \, \text{m/s}$; after 2 seconds, $v = 19.6 \, \text{m/s}$; and so on. The velocity increases linearly with time.

3. Consider an object thrown upward.

When you throw an object upward, it initially moves against gravity. Taking downward as positive, the initial velocity $u$ is negative. The equations become:

Better to use the vector form: $v = u + at$ where $u$ is negative and $a = +g$.

The object rises, slows down, reaches a maximum height where $v = 0$, then falls back down. The time to reach maximum height is found from $v = 0$:

Since $u$ is negative, $t$ is positive — which makes sense.

4. The independence of mass.

This is the most counterintuitive and important result. All objects in free fall — regardless of mass, size, or shape (in vacuum) — accelerate at the same rate $g$. Why?

From Newton's second law: $F = ma$. The gravitational force on an object is $F = mg$ (where $m$ is the gravitational mass). Setting these equal:

The mass cancels out! This is because gravitational mass and inertial mass are equivalent (the equivalence principle, which Einstein used as the foundation of general relativity).

5. Real-world considerations.

In practice, air resistance complicates free fall. A feather falls slower than a hammer because air drag opposes motion. The drag force depends on shape, cross-sectional area, and speed. For dense, streamlined objects (like a metal ball), air resistance is negligible over short distances, so free-fall equations work well.

Terminal velocity occurs when air drag equals weight, giving zero net acceleration. For a skydiver, terminal velocity is about $55 \, \text{m/s}$ (200 km/h) in a spread-eagle position, and higher when diving. …

Concept: Free Fall as a Consequence of $F=ma$

Method: Derive-from-First-Principles (force law $\rightarrow$ mass-independence $\rightarrow$ integrate)

Rather than starting from "objects fall with acceleration $g$" as a stated fact and then describing its consequences, this method derives that fact from Newton's second law, proves mass-independence explicitly, and only then integrates to recover the standard kinematic equations.

Steps

1. State the force acting on a freely falling body. Neglecting air resistance, the only force is gravity:

where $m$ is the object's (gravitational) mass and $g$ is the local gravitational field strength.

2. Apply Newton's second law to find the resulting acceleration:

3. Cancel $m$ from both sides. This step is the crux of free fall — the object's own mass appears on both sides of the equation (once as the source of the gravitational force $F=mg$, and once as the resistance to acceleration in $F=ma$), so it cancels exactly:

Every falling object, heavy or light, therefore has the same acceleration — independent of its mass. (This cancellation is only exact because gravitational mass and inertial mass are experimentally identical — the equivalence principle.)

4. Recover velocity by integrating acceleration. With $a=g$ constant and downward taken as positive, $a = \dfrac{dv}{dt} = g$:

5. Recover displacement by integrating velocity. $v = \dfrac{ds}{dt} = u+gt$: …