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# Gravity Time Equations for Falling Objects

by Ron Kurtus (revised 5 January 2011)

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 * time* it takes for a

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

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

## Time with respect to velocity

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

t = (v − v_{i})/g(

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:

t = v/g

where

**t**is the time in seconds**v**is the vertical velocity 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})

Since the object is moving in the direction of gravity, **v** is a positive number.

Elapsed time of a falling object as a function of velocity or displacement

## Time with respect to displacement

The general gravity equation for the elapsed time with respect to displacement is:

t = [ −v_{i }± √(v_{i}^{2}+ 2gy) ]/g

where

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

(

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

When the object is simply dropped, the initial velocity_{} is zero (**v _{i} =** 0) and the equation for elapsed time becomes:

t =_{ }± √(2gy)/g

Since time **t** is always positive, the equation is:

t = √(2gy)/g

Change **g** to **√(g ^{2} )** and simplify the equation:

t =_{ }√(2gy)/√(g^{2})

Thus, the resulting time equation is:

t =_{ }√(2y/g)

## Examples

The following examples illustrate applications of the equations.

### Given the velocity

How long does it take for a falling object to reach 224 ft/s?

#### Solution

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

t = v/g

t= (224 ft/s)/(32 ft/s^{2})

t= 7 seconds

### Given the displacement

How long does it take for an object to fall 200 meters?

#### Solution

Since displacement is in meters, **g** = 9.8 m/s^{2}. Substitute values in the equation:

t = √(2y/g)

t = √[2*(200 m)/(9.8 m/s^{2})]

t = √(40.8 s^{2})

t= 6.39 s

## Summary

There are simple equations for falling objects that allow you to calculate the time taken to achieve a given velocity or displacement. These equations are:

t = v/g

t = √(2y/g)

Check your numbers

## 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

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

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

## Questions and comments

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## Gravity Time Equations for Falling Objects