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 velocity. 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.

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_i²)

where

± means plus or minus
v is the vertical velocity in meters/second (m/s) or feet/second (ft/s)
√(2gy + v_i²) is the square root of the quantity (2gy + v_i²)
y is the vertical displacement in m or ft
g is the acceleration due to gravity (9.8 m/s² or 32 ft/s²)
v_i is the initial vertical velocity of the object

(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_i²)

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

v = √(2gy + v_i²)

Substitute values in the equation:

v = √[2*(9.8 m/s²)*(2 m) + (2 m/s)²]

v = √(39.2 m²/s² + 4 m²/s²)

v = √(43.2 m²/s²)

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

v = gt + v_i

Substitute values in the equation:

v = (9.8 m/s²)*(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: