# Gravitational force equation Physics

September 9, 2015

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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### Websites

- Physics Hypertextbook

- Wikipedia

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- Physics Classroom

### Books

Top-rated books on Simple Gravity Science

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