# Overview of Gravity Equations for Objects Projected Upward

by Ron Kurtus (revised 15 March 2018)

When an object is projected upward and released at some velocity, it is moving in the opposite direction of the force of gravity.

As the object moves upward, it slows down from this initial velocity until it reaches its peak or maximum height. Then the object falls toward the ground. By knowing the initial velocity, you can calculate the velocity, displacement and time of the object during its flight.

Note: Since our convention states that the direction of gravity is positive, the upward initial velocity is then a negative number. (See Convention for Direction in Gravity Equations for more information.)

It is assumed that the air resistance on the object is negligible. Also, there is a restriction on how high the object can be projected. At heights above 64 km or 40 mi, the value of the acceleration due to gravity changes enough to make your calculations inaccurate.

This lesson is an overview of the equations and has references to the other lessons, which provide the details.

Questions you may have include:

• What are the equations for velocity?
• What are the equations for displacement?
• What are the equations for time?

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

## Velocity equations

The equations for the velocity of an object projected upward at an initial velocity of vi are:

v = gt + vi

### With respect to displacement

v = −√(2gy + vi2) (going up)

vm = 0 (at maximum displacement)

v = +√(2gy + vi2) (coming down)

(See Gravity Velocity Equations for Objects Projected Upward for details.)

## Displacement equations

The equations for the displacement from the starting point of an object projected upward at an initial velocity of vi are:

### With respect to velocity

y = (v2 − vi2)/2g

ym = −vi2/2g (peak or maximum displacement)

### With respect to time

y = gt2/2 + vit

ym = −gtm2/2 (maximum displacement)

### Total distance traveled

du = |y| (going upward to maximum displacement)

dd = |2ym| + y (sum of going up and coming down)

(See Gravity Displacement Equations for Objects Projected Upward for details.)

## Time equations

The equations for the time an object projected upward at an initial velocity of vi travels are:

### With respect to velocity

t = (v − vi)/g

tm = −vi/g (time to reach maximum displacement)

### With respect to displacement

t = [−vi − √(vi2+ 2gy)]/g (going up)

tm = √(−2ym/g) (time to reach maximum displacement)

t = [−vi + √(vi2+ 2gy)]/g (coming down)

(See Gravity Time Equations for Objects Projected Upward for details.)

## Summary

When an object is projected upward and released at some initial velocity, it is moving in the opposite direction of the force of gravity, and the initial velocity is negative.

The object moves upward, reaches its peak or maximum displacement and then falls toward the ground. Knowing the initial velocity, you can calculate the velocity, displacement and time of the object during its flight.

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