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Overview of Gravity Equation Derivations

by Ron Kurtus (revised 22 April 2015)

Starting with the assumption that the acceleration due to gravity is a constant value, you can derive equations that define the relationships between velocity, displacement 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 displacement-time relationship is derived. Then the velocity-displacement equations are obtained from the previous two derivations.

Questions you may have include:

This lesson will answer those questions. Useful tool: Units Conversion



Overview of velocity-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

Note: Vectors have magnitude and direction and are indicated in boldface. Scalars have only magnitude and are in regular text.

(See Vectors in Gravity Equations for more information.)

Using Calculus, you integrate and derive the relationship between velocity and time for an object under the influence of gravity:

v = gt + vi

and

t = (v − vi)/g

where

See Derivation of Velocity-Time Gravity Equations for details of the derivations.

Overview of displacement-time relationships

The displacement a moving object travels in a given time is found by knowing that velocity is the change in displacement with respect to time:

v = dy/dt

Note: Displacement is a vector quantity denoting the change in position in a given direction, whereas distance is a scalar quantity that denotes the total change in position, independent of the path taken.

Substituting for v in the equation v = gt + vi and integrating, you get:

y = gt2/2 + vit

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 Displacement-Time Gravity Equations for details of the derivations.

Overview of displacement-velocity relationships

To determine the displacement required 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 Displacement-Velocity Gravity Equations for details of the derivations.

Summary

The relationships between the velocity of an object under the influence of gravity, the displacement 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.


Know where equations come from


Resources and references

Ron Kurtus' Credentials

Websites

Gravity Resources

Falling Bodies - Physics Hypertextbook

Equations for a falling body - Wikipedia

Gravity Calculations - Earth - Calculator

Books

Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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