Gravity Velocity Equations for Objects Projected Downward
About 2 min read
(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 + vi2)
where
- ± means plus or minus
- v is the vertical velocity in meters/second (m/s) or feet/second (ft/s)
- √(2gy + vi2) is the square root of the quantity (2gy + vi2)
- y is the vertical displacement in m or ft
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
- vi is the initial vertical velocity of the object
Velocity with respect to time
The general gravity equation for velocity with respect to time is:
v = gt + vi
where t is the time the object has fallen in seconds (s).
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 vi = 2 m/s and y = 2 m. Since vi is in m/s and y is in meters, then g = 9.8 m/s2. The equation to use is:
v = √(2gy + vi2)
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 vi = 10 m/s and t = 4 s. Since vi is in m/s, g = 9.8 m/s2. The equation to use is:
v = gt + vi
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:
v = √(2gy + vi2)
v = gt + vi
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