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Overview of Gravity Equation Derivations
by Ron Kurtus (revised 8 August 2010)
Starting with the assumption that the acceleration due to gravity is a constant value, you can derive equations that define the relationships between velocity, distance and time for an object moving under the influence of gravity. An initial velocity factor is included in these equations.
You use basic Calculus to determine the equations for the relationship between velocity and time for an object that is dropped, thrown downward or projected upward. From the velocity equation, the distance-time relationship is derived. Then the velocity-distance equations are obtained from the previous two derivations.
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.
Overview of velocity and time relationships
The velocity and time relationships as a result of the force of gravity are based on the fact that the acceleration due to gravity, g, is a constant value.
Since acceleration is the change in velocity with respect to time, the equation for the acceleration due to gravity is:
dv/dt = g
where
- dv is the derivative or small change in velocity
- dt is the derivative or increment of time
Using Calculus, you integrate and derive the relationship between velocity and time for a falling object:
v = gt + vi
and
t = (v − vi)/g
where
- v is the vertical velocity of the object in 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 (s)
- vi is the initial vertical velocity in m/s or ft/s
(See Derivation of Velocity-Time Gravity Equations for details of the derivations.)
Note: Since the force of gravity is downward, toward the ground, we use the convention that down is a positive direction. Some textbooks consider a downward velocity as a negative value.
Overview of 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 = dy/dt
Substituting for v in the equation v = gt + vi and integrating, you get:
y = gt2/2 + vit Note: Since our convention is that down is a positive direction, the downward distance y from the starting point is also positive.
Rearranging y = gt2/2 + vit and solving the quadratic equation for t gives you:
t = [ −vi± √(vi2+ 2gy) ]/g
This equation can create some confusion 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.)
Overview of 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 y = gt2/2 + vit from the previous derivations to get:
y = (v2 − vi2)/2g
Solving for v, you get:
v = ±√(2gy + 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 Gravity and Gravitation 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 Gravity Equation Derivations