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 velocity an object falls after a given period of time and the velocity it reaches at 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 velocity for a given time?
- What is the equation for the velocity to reach a given displacement?
- What are some examples of these equations?
Velocity with respect to time
The general gravity equation for velocity with respect to time is:
v = gt + vi
Since the initial velocity vi = 0 for an object that is simply falling, the equation reduces to:
v = gt
- v is the vertical velocity of the object 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)
- t is the time in seconds (s) that the object has fallen
Velocity of a falling object as a function of time or displacement
Velocity with respect to displacement
The general gravity equation for velocity with respect to displacement is:
v = ±√(2gy + vi2)
- ± means plus or minus
- √(2gy + vi2) is the square root of the quantity (2gy + vi2)
- y is the vertical displacement in meters (m) or feet (ft)
Since vi = 0, y is positive because it is below the starting point. Also, v is downward and positive. Only the + term of ± applies.
Thus, the equation for the velocity of a falling object after it has traveled a certain displacement is:
v = √(2gy)
The following examples illustrate applications of the equations.
For a given time
What will be the velocity of an object after it falls for 3 seconds?
Substitute in the equation:
If you use g = 9.8 m/s2, v = (9.8 m/s2)*(3 s) = 29.4 m/s.
If you use g = 32 ft/s2, v = (32 ft/s2)*(3 s) = 96 ft/s.
For a given displacement
What is the velocity of an object after it has fallen 100 feet?
Since y is in feet, g = 32 ft/s2. Substitute in the equation:
v = √[2*(32 ft/s2)*(100 ft)]
v = √(6400 ft2/s2)
v = 80 ft/s
There are simple equations for falling objects that allow you to calculate the velocity the object reaches for a given displacement or time. The equations are:
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