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Gravity Time Equations for Falling Objects
by Ron Kurtus (revised 5 January 2011)
When you drop an object from some height above the ground, it has an initial velocity of zero. Simple equations allow you to calculate the time it takes for a falling object to reach a given velocity and the time it takes to reach a given displacement. The equations assume that air resistance is negligible.
Examples demonstrate applications of the equations.
Questions you may have include:
- What is the equation for the time to reach a given velocity?
- What is the equation for the time to reach a given displacement?
- What are some examples of these equations?
This lesson will answer those questions.
Useful tools: Metric-English Conversion | Scientific Calculator.
Time with respect to velocity
The general gravity equation for elapsed time with respect to velocity is:
t = (v − vi)/g
(See Derivation of Velocity-Time Gravity Equations for details of the derivation.)
Since the initial velocity vi = 0 for an object that is simply falling, the equation reduces to:
t = v/g
where
- t is the time in seconds
- v is the vertical velocity in meters/second (m/s) or feet/second (ft/s)
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
Since the object is moving in the direction of gravity, v is a positive number.
Elapsed time of a falling object as a function of velocity or displacement
Time with respect to displacement
The general gravity equation for the elapsed time with respect to displacement is:
t = [ −vi ± √(vi2 + 2gy) ]/g
where
- ± means plus-or-minus
- √(vi2 + 2gy) is the square root of the quantity (vi2 + 2gy)
- y is the vertical displacement in meters (m) or feet (ft)
(See Derivation of Displacement-Time Gravity Equations for details of the derivation.)
When the object is simply dropped, the initial velocity is zero (vi = 0) and the equation for elapsed time becomes:
t = ± √(2gy)/g
Since time t is always positive, the equation is:
t = √(2gy)/g
Change g to √(g2 ) and simplify the equation:
t = √(2gy)/√(g2 )
Thus, the resulting time equation is:
t = √(2y/g)
Examples
The following examples illustrate applications of the equations.
Given the velocity
How long does it take for a falling object to reach 224 ft/s?
Solution
Since v is in ft/s, g = 32 ft/s2. Substitute values in the equation:
t = v/g
t = (224 ft/s)/(32 ft/s2)
t = 7 seconds
Given the displacement
How long does it take for an object to fall 200 meters?
Solution
Since displacement is in meters, g = 9.8 m/s2. Substitute values in the equation:
t = √(2y/g)
t = √[2*(200 m)/(9.8 m/s2)]
t = √(40.8 s2)
t = 6.39 s
Summary
There are simple equations for falling objects that allow you to calculate the time taken to achieve a given velocity or displacement. These equations are:
t = v/g
t = √(2y/g)
Check your numbers
Resources and references
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
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Gravity Time Equations for Falling Objects