Free Fall Formula

The Formula

v = v_0 + gt \quad ; \quad d = \frac{1}{2}gt^2

When to use: A dropped ball accelerates at the same rate regardless of its mass.

Quick Example

On Earth, objects fall with a = g \approx 9.8 \text{ m/s}^2 (or 10 \text{ m/s}^2 approximation).

Notation

g \approx 9.81 m/sĀ² is the acceleration due to gravity near Earth's surface, v_0 is the initial velocity, v is the velocity at time t, and y is the vertical position.

What This Formula Means

Motion under gravity alone, with no air resistance ā€” all objects in free fall accelerate at g \approx 9.81 m/sĀ² regardless of mass.

A dropped ball accelerates at the same rate regardless of its mass.

Formal View

In free fall near Earth's surface, \vec{a} = -g\hat{j} where g \approx 9.81 m/sĀ². The kinematic equations become y = y_0 + v_0 t - \frac{1}{2}gt^2 and v = v_0 - gt (taking upward as positive). The time to fall from height h from rest is t = \sqrt{2h/g}.

Worked Examples

Example 1

easy
A stone is dropped from a cliff. How far does it fall in 3 \text{ s}? Use g = 9.8 \text{ m/s}^2.

Solution

  1. 1
    Initial velocity: v_0 = 0 (dropped from rest).
  2. 2
    Use the free-fall displacement equation: d = v_0 t + \frac{1}{2}gt^2.
  3. 3
    d = 0 + \frac{1}{2}(9.8)(9) = 44.1 \text{ m}

Answer

d = 44.1 \text{ m}
In free fall, the only force acting is gravity. All objects fall at the same rate regardless of mass (ignoring air resistance), accelerating at g = 9.8 \text{ m/s}^2.

Example 2

medium
A ball is thrown upward with an initial velocity of 20 \text{ m/s}. How high does it go, and how long until it returns? Use g = 9.8 \text{ m/s}^2.

Common Mistakes

  • Thinking heavier objects fall faster ā€” in the absence of air resistance, all objects fall at the same rate; a feather and a hammer dropped on the Moon land together.
  • Forgetting that an object thrown upward is still in free fall the entire time ā€” gravity acts on it continuously, including at the very top where its velocity is momentarily zero.
  • Using the wrong sign for g ā€” if you define 'up' as positive, then g should be negative (-9.8 m/sĀ²); mixing up signs is the most common source of errors.

Why This Formula Matters

Free fall is the foundation for understanding projectile motion, orbital mechanics, and gravitational acceleration. It explains why astronauts float in the space station (they are in continuous free fall), how parachutes work, and how Galileo overturned centuries of wrong thinking about falling objects.

Frequently Asked Questions

What is the Free Fall formula?

Motion under gravity alone, with no air resistance ā€” all objects in free fall accelerate at g \approx 9.81 m/sĀ² regardless of mass.

How do you use the Free Fall formula?

A dropped ball accelerates at the same rate regardless of its mass.

What do the symbols mean in the Free Fall formula?

g \approx 9.81 m/sĀ² is the acceleration due to gravity near Earth's surface, v_0 is the initial velocity, v is the velocity at time t, and y is the vertical position.

Why is the Free Fall formula important in Physics?

Free fall is the foundation for understanding projectile motion, orbital mechanics, and gravitational acceleration. It explains why astronauts float in the space station (they are in continuous free fall), how parachutes work, and how Galileo overturned centuries of wrong thinking about falling objects.

What do students get wrong about Free Fall?

Heavier objects fall at the same rate as lighter ones (ignoring air).

What should I learn before the Free Fall formula?

Before studying the Free Fall formula, you should understand: acceleration.

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