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# Gravity Displacement Equations for Objects Projected Downward

by Ron Kurtus (updated 29 May 2023)

The displacement of an object is the change in position from the starting point in a specific direction and can be represented as a vector. It is different from distance, where direction is not indicated.

When you throw or * project an object downward*, it is accelerated until it is released at some velocity. If you know this initial velocity, there are simple derived

*that allow you to calculate the*

**equations***traveled from the starting point when the object reaches a given velocity or when it reaches a given elapsed time.*

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

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

## Displacement with respect to velocity

The general gravity equation applies in the case where you project the object downward and release it at an initial velocity ** v_{i}**. The result is that

**is a positive number, as are**

**v**_{i}**y**and

**v**:

y = (v^{2}− v_{i}^{2})/2g

where

**y**is the vertical displacement from the starting point in meters (m) or feet (ft)**v**is the vertical velocity in meters/second (m/s) or feet/second (ft/s)**v**is the initial vertical velocity in m/s or ft/s_{i}**g**is the acceleration due to gravity (9.8 m/s^{2}or 32 ft/s^{2})

(

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

Downward displacement as a function of velocity or time

## Displacement with respect to time

The equation of the displacement traveled within a given time for an object projected downward is:

y = gt+^{2}/2v_{i}t

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

(

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

Since the displacement is below the starting point, **y** is a positive number.

## Examples

The following examples illustrate applications of the equations.

### Displacement for a given velocity

If you throw an object downward at 10 m/s, find the minimum elevation from which you must throw the object so that it reaches 50 m/s.

#### Solution

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

**v**= 50 m/s. Since

**v**and

_{i}**v**are in m/s, then

**= 9.8 m/s**

g

g

^{2}. The equation to use is:

y = (v^{2}− v_{i}^{2})/2g

Substitute values in the equation:

y =[(50 m/s)^{2}− (10 m/s)^{2}]/2*(9.8 m/s^{2})

y= [(2500 m^{2}/s^{2}) − (100 m^{2}/s^{2})]/(19.6 m/s^{2})

y= (2400 m^{2}/s^{2})/(19.6 m/s^{2})

y= 122.4 m

### Displacement for a given time

If you throw an object downward at 30 ft/s and it travels for 4 seconds, find the displacement.

#### Solution

You are given that **v _{i}** = 30 ft/s and

**t**= 4 s. Since

**v**is in ft/s,

_{i}**g**= 32 ft/s

^{2}. The equation to use is:

y = gt+^{2}/2v_{i}t

Substitute values in the equation:

y =[(32 ft/s^{2})*(4 s)^{2}]/2 + (30 ft/s)*(4 s)

y =(32 ft/s^{2})*(16 s^{2})/2 + 120 ft

y =(512 ft)/2 + 120 ft

y =256 ft + 120 ft

y =376 ft

## Summary

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

y = (v^{2}− v_{i}^{2})/2g

y = gt+^{2}/2v_{i}t

Check your numbers

## 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_displacement.htm**

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## Gravity Displacement Equations for Objects Projected Downward