Overview of Gravity Equations for Objects Projected Upward

(updated 29 May 2023)

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.

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

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Velocity equations

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

With respect to time

v = gt + v_i

With respect to displacement

v = −√(2gy + v_i²) (going up)

v_m = 0 (at maximum displacement)

v = +√(2gy + v_i²) (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 v_i are:

With respect to velocity

y = (v² − v_i²)/2g

y_m = −v_i²/2g (peak or maximum displacement)

With respect to time

y = gt²/2 + v_it

y_m = −gt_m²/2 (maximum displacement)

Total distance traveled

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

d_d = |2y_m| + 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 v_i travels are:

With respect to velocity

t = (v − v_i)/g

t_m = −v_i/g (time to reach maximum displacement)

With respect to displacement

t = [−v_i − √(v_i²+ 2gy)]/g (going up)

t_m = √(−2y_m/g) (time to reach maximum displacement)

t = [−v_i + √(v_i²+ 2gy)]/g (coming down)

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

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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Seek understanding

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Resources and references

Websites

Gravity Resources

Gravity Calculations – Earth – Calculator