Free Fall Calculator

Use this free fall calculator & solver to find velocity, time, and distance for falling objects. Supports different gravitational values for Earth, Moon, Mars, and custom planets. Essential for physics problems involving gravity and projectile motion.

What is Free Fall?

Free fall is motion under the influence of gravity alone, with no air resistance. All objects in free fall near Earth’s surface accelerate downward at approximately \( 9.81 \text{ m/s}^2 \), regardless of their mass.

Galileo demonstrated this principle: a feather and hammer dropped in a vacuum hit the ground simultaneously.

Free Fall Equations

These are kinematic equations with ( a = g ) (gravitational acceleration):

Velocity After Time t

$$ v = v_0 + gt $$

Distance Fallen

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

For objects dropped from rest \( (v_0 = 0) \):

$$ h = \frac{1}{2}gt^2 $$

Velocity After Falling Height h

$$ v^2 = v_0^2 + 2gh $$

For objects dropped from rest:

$$ v = \sqrt{2gh} $$

Time to Fall Height h

For objects dropped from rest:

$$ t = \sqrt{\frac{2h}{g}} $$

Gravitational Acceleration Values

Sign Conventions

Two common conventions:

• Downward positive: \( g = +9.81 \text{ m/s}^2 \), heights decrease as you go up
• Upward positive: \( g = -9.81 \text{ m/s}^2 \), heights increase as you go up

This calculator uses downward as positive for falling objects.

Velocity vs Height Graph

For an object dropped from rest:

• Velocity increases linearly with time: \( v = gt \)
• Velocity relates to height by: \( v = \sqrt{2gh} \)

Common Calculations

Time to Fall a Given Height

From \( h = \frac{1}{2}gt^2 \):

$$ t = \sqrt{\frac{2h}{g}} $$

Example: An object dropped from 45 m

\( t = \sqrt{\frac{2 \times 45}{9.81}} = \sqrt{9.17} = 3.03 \) seconds

Impact Velocity

From \( v = \sqrt{2gh} \):

Example: Falling 45 m

\( v = \sqrt{2 \times 9.81 \times 45} = \sqrt{883} = 29.7 \) m/s (about 107 km/h)

Height for Given Fall Time

From \( h = \frac{1}{2}gt^2 \):

Example: 4 second fall

\( h = \frac{1}{2} \times 9.81 \times 16 = 78.5 \) m

Thrown Objects

Objects thrown vertically upward also experience free fall. At the highest point, velocity is zero, then the object falls back down.

Maximum Height (thrown upward)

$$h_{max} = \frac{v_0^2}{2g}$$

Total Time in Air (thrown upward, lands at same height)

$$ t_{total} = \frac{2v_0}{g} $$

Real-World Considerations

Air Resistance

Real falling objects experience air drag that:

• Increases with speed
• Depends on shape and size
• Eventually causes terminal velocity

Terminal Velocity

When air resistance equals gravitational force, acceleration stops. Typical values:

• Skydiver (spread): ~55 m/s (200 km/h)
• Skydiver (diving): ~90 m/s (320 km/h)
• Raindrop: ~9 m/s

Applications

• Calculating drop times and impact speeds
• Designing safety equipment
• Sports physics (diving, cliff jumping)
• Engineering drop tests
• Astronomical calculations

Example Problems

Problem 1: Dropped Object

A stone is dropped from a 100 m cliff. Find the time to hit the ground and impact velocity.

Time: \( t = \sqrt{\frac{2 \times 100}{9.81}} = 4.52 \) s

Velocity: \( v = 9.81 \times 4.52 = 44.3 \) m/s

Problem 2: Thrown Upward

A ball is thrown upward at 20 m/s. How high does it go?

\( h_{max} = \frac{20^2}{2 \times 9.81} = \frac{400}{19.62} = 20.4 \) m