Gravity Displacement Equations for Objects Projected Downward
About 3 min read
(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 vi. The result is that vi is a positive number, as are y and v:
y = (v2 − vi2)/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)
- viis 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)
Displacement with respect to time
The equation of the displacement traveled within a given time for an object projected downward is:
y = gt2/2 + vit
where t is the time the object has fallen in seconds (s).
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 vi = +10 m/s and v = 50 m/s. Since vi and v are in m/s, then
g = 9.8 m/s2. The equation to use is:
y = (v2 − vi2)/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 vi = 30 ft/s and t = 4 s. Since vi is in ft/s, g = 32 ft/s2. The equation to use is:
y = gt2/2 + vit
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:
y = (v2 − vi2)/2g
y = gt2/2 + vit
Check your numbers
Resources and references
Websites
Equations for a falling body – Wikipedia
Gravity Calculations – Earth – Calculator
Kinematic Equations and Free Fall – Physics Classroom