Free Fall Calculator

The Truth About Gravity

In physics class, you probably learned the standard formula for free fall: d = ½gt². While this equation is mathematically beautiful, it assumes you are falling in a vacuum (like on the Moon).

Here on Earth, we have an atmosphere. As an object falls through the air, the air pushes back. This opposing force is called Air Resistance (or Drag). For very short falls or very dense objects (like a bowling ball dropped 1 meter), drag is negligible. But for skydivers, raindrops, or anything falling from a significant height, air resistance changes everything.

What is Terminal Velocity?

Imagine a skydiver jumping out of a plane.
1. Gravity pulls them down, accelerating them faster and faster.
2. As they speed up, the Air Resistance pushing up against them increases.
3. Eventually, the upward force of Air Resistance exactly equals the downward force of Gravity.

At this precise moment, acceleration stops. The skydiver continues to fall, but their speed stays constant. This maximum speed is called Terminal Velocity.

Real World Myth: The Penny Drop

There is a popular urban legend that if you drop a penny from the top of the Empire State Building, it will accelerate enough to kill a pedestrian on the sidewalk below.

This is false. Why? Air Resistance.

A penny is light and has a flat shape that catches a lot of air (it tumbles). If we calculate it using the formula above, a penny reaches its terminal velocity after falling only about 15 meters. That speed is roughly 30 to 50 km/h (18-30 mph).

If it hit you, it would sting, but it wouldn't penetrate your skull. Physics saves the day!

Understanding the Inputs

To use our calculator accurately, you need to understand the variables:

1. Mass (m)

Heavier objects generally fall faster because they have more gravitational force to overcome the air resistance. A ping pong ball and a golf ball are the same size (Area), but the golf ball is heavier, so it has a higher terminal velocity.

2. Cross-Sectional Area (A)

This is the "shadow" the object casts if the sun were directly above it. A skydiver falling belly-down has a large Area (slow fall). If they dive head-first, they reduce their Area, drastically increasing their speed.

3. Drag Coefficient ($C_d$)

This is a dimensionless number that represents how aerodynamic a shape is.

• 0.47: A Sphere (Ball).
• 0.82: A Long Cylinder (Person standing up).
• 1.0 - 1.3: A Skydiver (Belly to earth, wearing baggy clothes).
• 0.04: A streamlined teardrop shape.

The Math Behind the Calculator

Calculating the time to fall with air resistance requires solving differential equations. The velocity at any given time $t$ is modeled by the hyperbolic tangent function ($\tanh$):

As time ($t$) increases, the $\tanh$ part gets closer and closer to 1, meaning the velocity $v(t)$ gets closer and closer to Terminal Velocity ($v_t$).

Skydiving Physics

A typical human skydiver (75kg) falling in a spread-eagle position reaches a terminal velocity of about 53 m/s (roughly 120 mph or 190 km/h). They usually hit this speed after falling for about 12 seconds (or 450 meters).

However, speed skydivers who tuck their arms and legs in to become more aerodynamic can reach speeds over 200 mph (320 km/h). The current world record (from a specialized jump from the stratosphere) is over 800 mph, breaking the sound barrier, because the air density ($\rho$) was so low at that altitude!