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 equations that allow you to calculate the displacement traveled from the starting point when the object reaches a given velocity or when it reaches a given elapsed time.

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 v_i is a positive number, as are y and v:

y = (v² − v_i²)/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_iis the initial vertical velocity in m/s or ft/s
g is the acceleration due to gravity (9.8 m/s² or 32 ft/s²)

(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 = gt²/2 + v_it

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_i and v are in m/s, then
g = 9.8 m/s². The equation to use is:

y = (v² − v_i²)/2g

Substitute values in the equation:

y = [(50 m/s)² − (10 m/s)²]/2*(9.8 m/s²)

y = [(2500 m²/s²) − (100 m²/s²)]/(19.6 m/s²)

y = (2400 m²/s²)/(19.6 m/s²)

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_i is in ft/s, g = 32 ft/s². The equation to use is:

y = gt²/2 + v_it

Substitute values in the equation:

y = [(32 ft/s²)*(4 s)²]/2 + (30 ft/s)*(4 s)

y = (32 ft/s²)*(16 s²)/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: