Free Fall Formula in Physics

What Is Free Fall?

Free fall is the motion of an object when gravity is the only force acting on it. No air resistance. No friction. Just the pull of Earth's gravity accelerating the object downward.

This is a simplified physics model. In reality, air resistance always exists. But for most introductory physics problems, you ignore it unless told otherwise. 🔑

The object can be moving upward, downward, or starting from rest—the key requirement is that gravity is the sole influence on its motion.

The Free Fall Formula

The primary equations you'll use:

s = ½gt²

Where:

• s = displacement (distance fallen), measured in meters
• g = acceleration due to gravity ≈ 9.8 m/s²
• t = time elapsed, measured in seconds

This formula works for objects dropped from rest. If the object has an initial velocity, you need a different version.

Breaking Down the Variables

Let's be clear about what each symbol means:

• s (displacement): How far the object has fallen from its starting point. Not the same as distance traveled if the object bounces or changes direction.
• g (gravity): On Earth, this is approximately 9.8 m/s². On the Moon, it's about 1.6 m/s². On Mars, roughly 3.7 m/s².
• t (time): Time since the object was released. This is always positive.

Acceleration Due to Gravity

Gravity accelerates everything at the same rate. A feather and a bowling ball fall at identical speeds in a vacuum. In air, the feather floats because air resistance fights gravity.

The value g = 9.8 m/s² means that every second, an object's downward velocity increases by 9.8 meters per second.

After 1 second: velocity = 9.8 m/s

After 2 seconds: velocity = 19.6 m/s

After 3 seconds: velocity = 29.4 m/s

You can calculate this with the velocity formula: v = gt

How to Solve Free Fall Problems

Step-by-Step Example

Problem: A ball is dropped from a 45-meter tall building. How long does it take to hit the ground?

Step 1: Identify what you know

• s = 45 m
• g = 9.8 m/s²
• Initial velocity (v₀) = 0 (dropped, not thrown)

Step 2: Pick the right formula

Use: s = v₀t + ½gt²

Since v₀ = 0, this simplifies to: s = ½gt²

Step 3: Plug in and solve

45 = ½(9.8)t²

45 = 4.9t²

t² = 45 ÷ 4.9

t² = 9.18

t = 3.03 seconds

That's your answer. Done.

Key Free Fall Equations

Depending on what's given and what's asked, you'll use different formulas:

The last two formulas include the initial velocity (v₀). If something is thrown upward, gravity works against the motion, hence the negative sign.

Common Mistakes to Avoid

• Using the wrong sign: If an object is thrown upward, displacement can be positive while velocity becomes negative. Don't mix these up.
• Confusing velocity with acceleration: Velocity changes. Acceleration is constant at 9.8 m/s². They're not the same thing.
• Forgetting g is positive: In the equation v = v₀ + gt, g is positive even though the object moves downward. The direction is handled by the signs of velocity and displacement.
• Using time incorrectly: Time is always positive. If your equation gives a negative time, you've got the wrong setup.

Real-World Applications

Free fall calculations matter in:

• Engineering: Designing parachutes, safety equipment, and drop tests
• Forensics: Determining fall heights from injury patterns
• Sports: Analyzing dives, ski jumps, and bungee cord physics
• Space science: Calculating orbital mechanics and re-entry trajectories

When a skydiver jumps from a plane, they're in free fall—until they hit terminal velocity when air resistance balances gravity. Before that point, the free fall formulas apply.

Quick Reference

Remember these core values:

• Gravity on Earth: 9.8 m/s²
• Basic distance formula: s = ½gt²
• Velocity from gravity: v = gt
• Velocity without time: v² = 2gs

Master these four equations and you can solve any basic free fall problem in physics.