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# Gravity Velocity Equations for Objects Projected Downward

by Ron Kurtus (updated 29 May 2023)

When you throw or * project an object downward*, it is accelerated until it is released at some

*. If you know this initial velocity, there are simple derived equations that allow you to calculate the velocity when the object reaches a given displacement from the starting point or when it reaches a given elapsed time.*

**velocity**Examples illustrate these equations.

Note: You normally do not need to memorize these equations, but you should know where to find them in order to solve equations.

Questions you may have include:

- How do you find the velocity for a given displacement?
- How do you find the velocity for a given time?
- What are some examples of these equations?

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

## 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**v**is the vertical velocity in meters/second (m/s) or feet/second (ft/s)**√(2gy +**is the square root of the quantity**v**)_{i}^{2}**(2gy +****v**)_{i}^{2}**y**is the vertical displacement in m or ft**g**is the acceleration due to gravity (9.8 m/s^{2}or 32 ft/s^{2})**v**is the initial vertical velocity of the object_{i}

(

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

Since **v** is a downward vector, it has a positive value. Likewise, **y** and ** v_{i}** are positive numbers. Thus, only the

**+**version of the equation applies:

v = √(2gy +v)_{i}^{2}

Velocity of object projected downward as a function of displacement or time

## Velocity with respect to time

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

v = gt + v_{i}

where **t** is the time the object has fallen in seconds (s).

(

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

This same equation applies for an object projected downward.

## Examples

The following examples illustrate applications of the equations.

### For a given displacement

Find the velocity of a rock that is thrown down at 2 m/s after it has traveled 2 meters.

#### Solution

You are given that **v _{i}** = 2 m/s and

**y**= 2 m. Since

**v**is in m/s and

_{i}**y**is in meters, then

**g**= 9.8 m/s

^{2}. The equation to use is:

v = √(2gy +v)_{i}^{2}

Substitute values in the equation:

v = √[2*(9.8 m/s^{2})*(2 m) + (2 m/s)^{2}]

v = √(39.2 m^{2}/s^{2}+ 4 m^{2}/s^{2})

v = √(43.2 m^{2}/s^{2})

v= 6.57 m/s

### For a given time

Suppose you throw the object downward at 10 m/s. Find its velocity after 4 seconds.

#### Solution

You are given that **v _{i}** = 10 m/s and

**t**= 4 s. Since

**v**is in m/s,

_{i}**g**= 9.8 m/s

^{2}. The equation to use is:

v = gt + v_{i}

Substitute values in the equation:

v= (9.8 m/s^{2})*(4 s) + 10 m/s

v= 39.2 m/s + 10 m/s

v= 49.2 m/s

## Summary

You can calculate the velocity when an object that is projected downward reaches a given displacement from the starting point or when it reaches a given elapsed time from the equations:

v = √(2gy +v)_{i}^{2}

v = gt + v_{i}

Help other people learn

## Resources and references

### Websites

**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_downward_velocity.htm**

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## Gravity Velocity Equations for Objects Projected Downward