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Explanation of Overview of Derivation of Gravity Equations - Succeed in Understanding Physics. Also refer to physical science, falling objects, Calculus, acceleration, velocity, integral, derivative, integrate, distance, time, relationships, Ron Kurtus, School for Champions. Copyright © Restrictions
Overview of Derivation of Gravity Equations
by Ron Kurtus (revised 30 August 2009)
Gravity is defined as the gravitational force on or near the surface of the Earth. The equation for the force of gravity is F = mg, where g is the acceleration due to gravity and is a constant value.
Using the fact that acceleration is a constant, you can derive the equations for the velocity, distance and time an object travels from its starting point.
You can use basic Calculus to first derive the equation for the velocity of an object that is dropped, thrown downward or projected upward after a given time, as well as the time for a given velocity. Then, you can derive the distance-time relationship from the velocity equation. The velocity-distance equations can then be obtained from the previous two derivations.
Advanced Physics and Physical Science students are often required to derive equations. Beginning students who have not yet taken Calculus can skip or skim this lesson.
Questions you may have include:
- What are the velocity and time relationships?
- What are the distance and time relationships?
- What are the velocity and distance relationships?
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 relationships
The starting point for the derivation of gravity equations comes from the equation of the force of gravity F = mg, where g is a constant equal to 9.8 m/s2 or 32 ft/s2.
Thus, you have acceleration, which is the change in velocity with respect to time, is a constant: a = g.
dv/dt = g
where
- dv is the derivative or small change in velocity
- dt is the derivative or increment of time
Using Calculus, you can integrate and derive the relationship between velocity and time for a falling object.
v = gt + vi
t = (v − vi)/g
where
- v is the velocity of the object (m/s or ft/s)
- g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
- t is the time in seconds
- vi is the initial velocity
(See Derivation of Velocity-Time Gravity Equations for details of the derivations.)
Distance and time relationships
The distance a falling object travels in a given time is found by knowing that velocity is the change in distance with respect to time:
v = dx/dt
Substituting for v in the equation from the previous derivation v = gt + vi and integrating, you get:
x = gt2/2 + vit
Rearranging x = gt2/2 + vit and solving the quadratic equation for t gives you:
t = [ −vi± √(vi2+ 2gx) ]/g
This equation creates some problems because of the plus-or-minus sign. If the object is thrown downward, the plus (+) sign is used. If the object is thrown upward, the sign depends on the object's position with respect to the starting point.
(See Derivation of Distance-Time Gravity Equations for details of the derivations.)
Velocity and distance relationships
To determine the distance the object travels to reach a given velocity, start with the equations t = (v − vi)/g and x = gt2/2 + vit from the previous derivations to get:
x = (v2 − vi2)/2g
Solving for v, you get:
v = ±√(2gx + vi2)
(See Derivation of Velocity-Distance Gravity Equations for details of the derivations.)
Summary
The relationships between the velocity of a falling object, the distance traveled and the time it takes to fall start with a basic equation a = g. You use calculus to integrate equations, use algebra for substitutions and perform other operations to get the results. Details of the derivations are seen at:
- Derivation of Velocity-Time Gravity Equations
- Derivation of Distance-Time Gravity Equations
- Derivation of Velocity-Distance Gravity Equations
See the Side Menu for more Gravitation and Gravity topics
Know where equations come from
Resources
The following resources provide information on this subject:
Websites
Acceleration due to Gravity Calculations - from Western Washington University
Gravity and Gravitation Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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Overview of Derivation of Gravity Equations