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# Velocity Equations for Falling Objects

by Ron Kurtus (updated 29 May 2023)

When you drop an object from some height above the ground, it has an initial velocity of zero. Simple * equations* allow you to calculate the

*a*

**velocity***reaches after a given period of time and its velocity at a given displacement. The equations assume that air resistance is negligible.*

**falling object**Examples demonstrate applications of the equations.

Questions you may have include:

- What is the equation for the velocity for a given time?
- What is the equation for the velocity to reach a given displacement?
- What are some examples of these equations?

This lesson will answer those questions. Useful tool: Units Conversion

## Velocity with respect to time

The general gravity equation for velocity with respect to time is:

v = gt + v_{i}(

See Derivation of Velocity-Time Gravity Equations for details of the derivation.)

Since the initial velocity **v _{i} =** 0 for an object that is simply falling, the equation reduces to:

v = gt

where

**v**is the vertical velocity of the object in meters/second (m/s) or feet/second (ft/s)**g**is the acceleration due to gravity (9.8 m/s^{2}or 32 ft/s^{2})**t**is the time in seconds (s) that the object has fallen

Velocity of a falling object as a function of time or displacement

## Velocity with respect to displacement

The general gravity equation for velocity with respect to displacement is:

v = ±√(2gy + v_{i}^{2})

where

**±**means plus or minus**√(2gy + v**is the square root of the quantity_{i}^{2})**(2gy + v**_{i}^{2})**y**is the vertical displacement in meters (m) or feet (ft)

(

See Derivation of Displacement-Velocity Gravity Equations for details of the derivations.)

Since **v _{i} =** 0,

**y**is positive because it is below the starting point. Also,

**v**is downward and positive. Only the

**+**term of

**±**applies.

Thus, the equation for the velocity of a falling object after it has traveled a certain displacement is:

v = √(2gy)

## Examples

The following examples illustrate applications of the equations.

### For a given time

What will be the velocity of an object after it falls for 3 seconds?

#### Solution

Substitute in the equation:

v = gt

If you use **g** = 9.8 m/s^{2}, **v** = (9.8 m/s^{2})*(3 s) = 29.4 m/s.

If you use **g** = 32 ft/s^{2}, **v** = (32 ft/s^{2})*(3 s) = 96 ft/s.

### For a given displacement

What is the velocity of an object after it has fallen 100 feet?

#### Solution

Since **y** is in feet, **g** = 32 ft/s^{2}. Substitute in the equation:

v = √(2gy)

v = √[2*(32 ft/s^{2})*(100 ft)]

v = √(6400 ft^{2}/s^{2})

v= 80 ft/s

## Summary

There are simple equations for falling objects that allow you to calculate the velocity the object reaches for a given displacement or time. The equations are:

v = gt

v = √(2gy)

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## Resources and references

### Websites

**Falling Bodies** - Physics Hypertextbook

**Equations for a falling body** - Wikipedia

**Gravity Calculations - Earth** - Calculator

**Kinematic Equations and Free Fall** - Physics Classroom

### Books

(Notice: The *School for Champions* may earn commissions from book purchases)

**Top-rated books on Simple Gravity Science**

**Top-rated books on Advanced Gravity Physics**

## Students and researchers

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gravity_equations_falling_velocity.htm**

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## Gravity Velocity Equations for Falling Objects