Overview of Gravity Equation Derivations

by Ron Kurtus (updated 29 May 2023)

You can derive the gravity equations for the relationship between velocity-time, displacement-time, and displacement-velocity of an object moving in the vertical direction.

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:

What are the velocity and time relationships?
What are the displacement and time relationships?
What are the velocity and displacement relationships?

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

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

dv is the derivative or small change in velocity
dt is the derivative or increment of time

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 + v_i

and

t = (v − v_i)/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)
v_i is the initial vertical velocity in m/s or ft/s

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

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 + v_i and integrating, you get:

y = gt²/2 + v_it

Rearranging y = gt²/2 + v_it and solving the quadratic equation for t gives you:

t = [−v_i ± √(v_i²+ 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.

Displacement-velocity relationships

To determine the displacement required to reach a given velocity, start with the equations
t = (v − v_i)/g and y = gt²/2 + v_it from the previous derivations to get:

y = (v² − v_i²)/2g

Solving for v, you get:

v = ±√(2gy + v_i²)

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