Explanation of Gravity Equations for Falling Objects - Succeed in Physical Science. Also refer to physics, force, mass, gravitational constant, acceleration, velocity, distance, time, calculation, square, square-root, Ron Kurtus, School for Champions. Copyright © Restrictions
Gravity Equations for Falling Objects
by Ron Kurtus (revised 10 October 2008)
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. (See Derivation of Gravity Equations for more information.)
Questions you may have include:
- What are the velocity equations?
- What are the distance equations?
- 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 equations
If you drop an object from some height, it will fall with increasing velocity.
Note: (We use velocity instead of speed, since the object is moving in a specific direction: down.
Velocity for a given time
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 light weight or not heavy will probably be affected by air resistance.
Example
If you drop an object from a tall building, you can calculate its velocity after 4 seconds:
v = gt
v = (9.8 meters/second-squared)*(4 seconds) = 39.2 meters/second
v = (32 ft/s²)*(4 s) = 128 ft/s
Velocity for a given distance
The equation for the velocity of a falling object after it has fallen a certain distance is:
v = SQRT(2gx) or v = √(2gx)
where
- SQRT and √ stand for square root
- √(2gx) is the square root of the quantity 2gx.
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
Find the velocity of a rock after it has fallen 4 feet.
v = √(2gx)
v = √(2 * 32 ft/s2 * 4 ft) = √(256 ft2/s2)
v = 16 feet/second
Distance equations
You can find the distance traveled for a given time and to reach a velocity.
Distance for a given time
The equation for the distance that the object falls in a given time is:
x = gt2/2
Example
Find the distance in meters that an object will fall in 3 seconds.
x = gt2/2
x = (9.8 m/s2) * (3 s) * (3 s) / 2
x = 44.1 meters
Distance for given velocity
The equation for the distance traveled to reach a given velocity is:
x = v2/2g
Example
Find the distance required to reach 100 miles per hour (mph).
First, you convert 100 mph to 146.7 feet/second. Then use that velocity in the equation.
x = v2/2g
x = (146.7 ft/s) * (146.6 ft/s) / (2 * 32 ft/s2)
x = 336.1 feet
That means that in order to get a ball to reach 100 mph, you need to go up to over 336 feet, which is about the height of a 34-story building.
Time equations
You can find the time it takes to reach a given velocity or distance.
Time to reach velocity
The equation for the time it takes to reach a given velocity is:
t = v/g
Example
Calculate Find the time it takes a falling object to reach a velocity of 49 m/s.
t = (49 m/s) / (9.8 m/s²)
t = 5 seconds.
Time to fall distance
The equation to determine the time it takes to fall a given distance is:
t = √(2x/g)
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)
t = √[2 * (64 ft) / ( 32 ft/s2)] = √(4 s2) = 2 seconds
Summary
The simple equations for falling objects are: v = gt, v = √(2gx), 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.
Check your numbers
Resources
The following resources provide information on this subject:
Websites
Acceleration of Gravity Calculations - from Western Washington University
Books
Top-rated books on Simple Gravity Science
Top-rated books on Advanced Gravity Physics
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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