Derivation of Displacement-Velocity Gravity Equations

(updated 29 May 2023)

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

Note: Displacement is a vector quantity denoting the change in position in a given direction. Velocity is a vector indicating the rate of change of position in a given direction.

See Derivation of Velocity-Time Gravity Equations and Derivation of Displacement-Time Gravity Equations lessons.

By substituting the time relationship in the displacement equation, you can determine the displacement with respect to velocity. From the derived displacement equation, you can then determine the equation for the velocity when the object reaches a given displacement 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.

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

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Basis for displacement-velocity derivations

To determine the displacement from the starting point at a given velocity, start with the equations:

t = (v − v_i)/g

(Obtained from Derivation of Velocity-Time Gravity Equations)

and

y = gt²/2 + v_it

(Obtained from Derivation of Displacement-Time Gravity Equations)

where

• t is the time in seconds (s)
• v is the vertical velocity in m/s or ft/s
• v_i is the initial vertical velocity in m/s or ft/s
• g is the acceleration due to gravity (9.8 m/s² or 32 ft/s²)
• y is the vertical displacement from the starting point in m or ft

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.

By substituting the equation for t in the equation for y, you can get the displacement with respect to velocity. Then by solving that equation for v, you get the velocity with respect to displacement equation.

Derivation of displacement for a given velocity

To derive the displacement equation, you can start with the time equation:

t = (v − v_i)/g

Square both sides of the equation:

t² = (v − v_i)²/g²

t² = (v² − 2vv_i + v_i²)/g²

Consider the displacement equation:

y = gt²/2 + v_it

Substitute for t² and t from the above equations:

y = g (v² − 2vv_i + v_i²)/2g² + v_i(v − v_i)/g

Multiply v_i(v − v_i)/g by 2/2 and combine like terms:

y = (v² − 2vv_i + v_i²)/2g + 2(vv_i − v_i²)/2g

y = (v² − 2vv_i + v_i² + 2vv_i − 2v_i²)/2g

The resulting general gravity equation for displacement with respect to velocity is:

y = (v2 − vi2)/2g

Derivation of velocity for a given displacement

To get the velocity for a given displacement, multiply both sides of
y = (v² − v_i²)/2g by 2g and solve for v:

2gy = (v² − v_i²)

v² = 2gy + v_i²

Take the square root of both sides of the equation to get the general gravity equation for velocity with respect to displacement:

v = ±√(2gy + vi2)

where

• ± means plus or minus
• √(2gy + v_i²) is the square root of the quantity (2gy + v_i²)

The derived equations are:

y = (v² − v_i²)/2g

v = ±√(2gy + v_i²)

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Be observant and curious

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Resources and references

Websites

Gravity Resources

Falling Bodies – Physics Hypertextbook

Equations for a falling body – Wikipedia

Gravity Calculations – Earth – Calculator

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