Explanation of Gravity Equations for Upward Motion - Succeed in Physical Science. Also refer to physics, initial velocity, project, throw, object, ball, gravitational constant, acceleration, distance, height, time, calculation, square, square-root, Ron Kurtus, School for Champions. Copyright © Restrictions
Gravity Equations for Upward Motion
by Ron Kurtus (23 August 2007)
When you throw or project an object upward, it slows down from its initial velocity until it reaches its peak. Then it falls back down to the ground. Given that initial velocity, you can calculate the velocity of the object at any time, as well as the time it takes to reach its peak. You can also determine how high the object will go. An exception is if the object is projected at such a high velocity that it will go off into outer space. That is called the escape velocity.
Questions you may have include:
- How long does it take an object to reach its peak?
- How do you find the maximum height of a projected object?
- How do you determine the escape velocity?
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 and time
When you throw an object upward, it starts at some initial velocity and slows down from the force of gravity until its velocity becomes zero (0). Then it reverses direction and starts falling back to earth.
You can find the velocity as it moves upward and the time it takes to reach its peak by applying the equation for falling objects: v = gt.
(See Gravity Equations for Falling Objects for more information.)
Velocity at any time
The velocity at any time (vt) is the initial velocity (vo) minus the velocity you would get from simply dropping the object (v). Thus, the velocity at any time is: vt = vo - v. Since the velocity of a falling object is v = gt, the equation for the velocity of an upward moving object at a given time is:
vt = vo - gt
where
- vt is the velocity after a given time
- vo is the initial velocity
- g is the gravitational acceleration constant 9.8 m/s² or 32 ft/s²
- t is the time measured
- gt is the product of g times t
Example
For example, if initial velocity that the object was projected upward was 49 m/s, the velocity after 3 seconds would be:
vt = vo - gt = 49 m/s - (3 s)*(9.8 m/s²) = 49 - 29.4 = 19.6 m/s
Time to reach the peak
At the peak of the upward flight, the velocity of the object becomes zero, since it stops and will start to descend. If vt = 0, then vo = gt, and the time to reach that peak is:
t = vo/g
Example
Thus, if a ball was thrown upward at 49 m/s, the time it would take to it reach its peak would be:
t = vo/g = (49 m/s) / (9.8 m/s²) = 5 seconds.
Maximum height
At normal initial velocities, an object projected upward will achieve some maximum height before it starts falling back to the ground.
An interesting fact is that if you drop the object from this maximum height, it will accelerate and reach the initial velocity at the point it hits the ground or the height from which it was thrown. That means that if you project an object upward from a height of 3 meters at some initial velocity, it will go up, reach its peak and then fall down, such that the velocity is the same as the initial velocity at that 3 meter height.
Equation
That fact can be used to determine the maximum height. We use the equation of the height necessary to reach a given velocity for a falling object: x = v²/2g, to get the equation:
xmax = vo²/2g
where
- xmax is the maximum height the object reaches
- vo is the initial velocity upward of the object and vo² is its square
- g is the constant acceleration due to gravity
Example
Suppose you threw a ball upward at 60 miles per hour (mph). Since g is in feet/second-squared, you need to convert miles into feet and hours into seconds in order to calculate the height it would go in feet. To convert 30 mph into feet per second, you can use the conversion calculator or know that 1 mile = 5280 feet and 1 hour = (60 * 60) = 3600 seconds. Thus, 60 mph = 60 * 5280 / 3600 = 88 feet/second. Thus:
xmax = vo²/2g = (88 ft/s)²/ [2 * (32 ft/s²)] = 88 * 88 / 64 = 126 feet.
Escape velocity
If you project the object upward at a sufficient velocity, it can actually escape from the gravity of the Earth and fly off into space. The derivation of the escape velocity is complex and takes into account the Earth's radius. But the equation is relatively simple:
ve = √(2gR)
where
- ve is the escape velocity
- g is acceleration due to gravity constant
- R is the radius of the Earth
- √(2gR) is the square root of (2gR); it is also written as SQRT(2gR)
Example
Since g = 9.8 m/s² and the radius of the Earth is approximately R = 6400 km, the escape velocity from the Earth is:
ve = SQRT(2 * 9.8 * 6400000) = 11,200 m/s. That is about 25,200 mph.
This equation can be used for any planet or sun, using the g and R for that particular heavenly body.
Summary
The velocity of an object projected upward at a given time is vt = vo - gt. If the initial velocity is not too great, the time it takes to reach its peak is t = vo/g and the maximum height the object would travel is xmax = vo²/2g. The initial velocity required for an object to escape the gravitational pull of the Earth is ve = √(2gR).
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. If a missile was fired upward at 11,200 m/s, how fast would it be going after 10 seconds?
2. What is the maximum height of a ball thrown up at 100 m/s?
3. If the Moon's radius is 1080 miles and its gravity is 1/6 of Earth's, what is its escape velocity?
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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Gravity Equations for Upward Motion