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Gravity Equations for Falling Objects

by Ron Kurtus (23 August 2007)

There are simple equations for falling objects that allow you to calculate the velocity and distance traveled, as well as the time taken to achieve a given velocity or distance.

These equations are based on the fact that the force of gravity for objects relatively close to Earth equals the mass of the object times the acceleration due to gravity (F = mg). The acceleration due to gravity (g) is constant for all objects up to altitudes beyond many space satellites. From this simple equation, it is determined that the velocity, distance and time are independent of the mass of the object, provided air resistance is negligible.

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:

• What is the velocity equation?
• What is the distance equation?
• What are the time equations?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

Useful tools: Metric-English Conversion | Scientific Calculator.

Velocity

If you drop an object from some height, it will fall with increasing velocity. (We use velocity instead of speed, since the object is moving in a specific direction: down.) You can determine the object's velocity according to the time it has fallen by the equation:

v = gt

where

• v is the velocity of the object
• g is the gravitational constant 9.8 m/s² or 32 ft/s²
• t is the measure time of the fall
• gt is the product of g times t

Note that the assumption is that air resistance on the object is negligible. An object that is lightweight or not heavy will probably be affected by air resistance.

Example

Thus, if you drop an object from a tall building, its velocity after 4 seconds, it will be:

v = gt = (4 seconds) * (9.8 meters/second-squared) = 39.2 meters/second

v = gt = (4 s) * (32 ft/s²) = 128 ft/s

Distance in fall

You can find the distance traveled for a given time and to reach a velocity.

Distance for a given time

Using Calculus, you can derive the equation for the distance that the object falls in a given time. This equation is

x = gt²/2

where

• x is the distance the object falls
• g is the acceleration due to gravity
• t is the time considered
• t² is t times t or t-squared
• gt²/2 is g times t-squared divided by 2

Example

The distance in meters that an object will fall in 3 seconds is:

x = gt²/2 = (9.8 m/s²) * (3 s) * (3 s) / 2 = 44.1 m

Reaching given velocity

By substituting t = v/g and v = gt in the x = gt²/2 equation, you can get the distance traveled to reach a given velocity: x = v²/2g

Example

For example, to find the distance required to reach 100 miles per hour (mph), you convert 100 mph to 146.7 feet/second. Thus:

x = v²/2g = (146.7) * (146.6) / (2 * 32) = 336.1 feet

That means you need to go up to over 336 feet to have a ball that you drop reach 100 mph. That is about the height of a 34-story building.

Time to reach value

You can find the time it takes to reach a given velocity or distance.

Time to reach velocity

To determine the time it takes to reach a given velocity, you can rearrange v = gt into:

t = v/g

Example

Thus, the time it takes a falling object to reach a velocity of 49 m/s is t = (49 m/s) / (9.8 m/s²) = 5 seconds.

Time to fall distance

To determine the time it takes to fall a given distance, you can rearrange the equation x = gt²/2 into:

t = SQRT(2x/g)
or
t = √(2x/g)

where SQRT and √ mean the square root of the quantity in the parentheses (2x/g).

Note: The definition of square root can be difficult to understand. The square root of some number X is the number that when multiplied by itself equals X. In other words, the square root of 9 is 3, because 3 times 3 equals 9. Most numbers require a calculator to determine their square root.

Example

If you dropped a weight from a height of 64 feet, you could calculate how long it takes to hit the ground.

t = √(2x/g) = √[2 * (64 ft) / ( 32 ft/s²)] = √(4 s²) = 2 seconds

Substituting t = 2 seconds into v = gt, you get a velocity of 64 feet/second.

Summary

Simple equations for falling objects allow you to calculate the velocity and distance traveled, as well as the time taken to achieve a given velocity or distance. These equations are based on force of gravity equation and the fact that the acceleration due to gravity (g) is constant for all objects up to altitudes beyond many space satellites.

The equations are: v = gt, x = gt²/2, x = v²/2g, t = v/g and t = SQRT(2x/g). Typically, you don't need to memorize them but know where to find them in order to solve problems.

Answers to Readers' Questions

Check your numbers

Resources

The following resources provide information on this subject:

Websites

Acceleration of Gravity Calculations - from Western Washington University

Physical Science Resources

Books

Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics

Miscellaneous

Mini-quiz to check your understanding

1. What is the velocity in meters/second of an object that has been falling for 10 seconds?

2. How far in meters can an object fall in 10 seconds?

3. How long does it take an object weighing 100 pounds to fall 144 feet?

If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.

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