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Derivation of Velocity-Time Gravity Equations

by Ron Kurtus (revised 22 April 2015)

The basis for the derivations of the velocity-time gravity equations starts with the assumption that the acceleration due to gravity is a constant value.

Since acceleration is also the change in velocity for an increment of time, you use Calculus to integrate that change to get the velocity for a given elapsed time. From the velocity equation, you can then determine the equation for the time it takes for the object to reach a given velocity from the starting point.

The derived equations are affected by the initial velocity of the object. This is important in later applications of the equations.

Questions you may have include:

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



Basis for velocity-time derivations

The derivations start with the assumption that the acceleration due to gravity g is a constant for displacements relatively close to Earth.

Acceleration is also the incremental change in velocity with respect to time:

a = dv/dt

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.)

Since g is the acceleration due to gravity:

a = g

and

dv/dt = g

Multiply both sides of the equation by dt to get:

dv = g*dt

By using Calculus to integrate this equation, you can get the equations for velocity and time.

Velocity-time relationship

Velocity-time relationship

Derivation of velocity for a given time

Integrate dv = g*dt on both sides of the equal sign.

First, integrate dv over the interval from v = vi to v = v:

∫dv = v − vi

where

Note: The initial velocity is the velocity at which the object is released after being accelerated from zero velocity. Initial velocity does not occur instantaneously.

Then, integrate g*dt over the time interval from t = 0 to t = t:

∫g*dt = gt − 0

The result of the two integrations is:

v − vi = gt

Thus, the general gravity equation for velocity with respect to time is:

v = gt + vi

Derivation of time for a given velocity

The time it takes to reach a given velocity is obtained by rearranging the equation v = gt + vi and solving for t:

v − vi = gt

t = (v − vi)/g

Summary

Starting with the fact that the acceleration due to gravity g is considered a constant and knowing that acceleration is the change in velocity for a change in time, you can derive the gravity equations for the velocity with respect to time. You can then determine the equation for the time to reach a given velocity.

The derived equations are:

v = gt + vi

t = (v − vi)/g


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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