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# Velocity Equations for Falling Objects

by Ron Kurtus

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)

Be a champion

## 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**

## Questions and comments

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## Gravity Velocity Equations for Falling Objects