# Derivation of Velocity-Time Gravity Equations

by Ron Kurtus (revised 15 March 2018)

You can derive the general gravity equations for the velocity of a falling object over a given time, as well as for the time it takes to reach a given velocity. The equations include the initial velocity of the object.

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:

• What is the basis for the derivations?
• What is the velocity for a given time equation?
• What is the time for a given velocity equation?

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. Also, acceleration is defined as the incremental change in velocity with respect to time:

a = dv/dt

where

• a is the acceleration
• dv is the first derivative of velocity v (a small change in velocity)
• dt is the first derivative of time t (a small time increment) Velocity-time relationship

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, as seen below.

## 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

• is the integral sign, as used in Calculus
• v is the vertical velocity of the object
• vi is the initial vertical velocity of the object

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

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Falling Bodies - Physics Hypertextbook

Equations for a falling body - Wikipedia

Gravity Calculations - Earth - Calculator

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