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

where

- 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)

### Examples

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?

#### Solution

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

### Summary

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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