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# Gravity Time Equations for Objects Projected Downward

by Ron Kurtus (revised 15 March 2018)

When you throw or * project an object downward*, it is accelerated until it is released at some velocity. If you know this initial velocity, there are simple derived equations that allow you to calculate the

*it takes for it to reach a given velocity or when it reaches a given displacement from the starting point.*

**time**Examples illustrate these equations.

Note: You normally do not need to memorize these equations, but you should know where to find them in order to solve equations.

Questions you may have include:

- How do you find the time for a given velocity?
- How do you find the time for a given displacement?
- What are some examples of these equations?

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

## Time with respect to velocity

The equation for the time it takes an object that is thrown or projected downward to reach a given velocity is:

t = (v − v_{i})/g

where

**t**is the time in seconds (s)**v**is the vertical velocity of the falling object in feet/second (ft/s) or meters/second (m/s)**v**is the initial vertical velocity the object has been projected downward in ft/s or m/s_{i}**g**is the acceleration due to gravity;**g**= 32 ft/s^{2}or 9.8 m/s^{2}

(

See the Derivation of Velocity-Time Gravity Equations lesson for details of the derivation.)

Since the object is moving in the direction of gravity, **v** and ** v_{i}** are positive numbers.

## Time with respect to displacement

The general gravity equation for the time with respect to displacement is:

t = [−v_{i}± √(v_{i}^{2}+ 2gy)]/g

where

**±**means plus or minus**y**is the vertical displacement in feet (ft) or meters (m)**√(v**is the square root of the quantity_{i}^{2}+ 2gy)**(v**_{i}^{2}+ 2gy)

(

See Derivation of Displacement-Time Gravity Equations for details of the derivation.)

Since **v _{i}** is downward, it has a positive value and

**−v**is obviously negative. This means that the

_{i}**+**version of the equation must be used in order to make

**t**a positive number. The equation is then:

t = [−v_{i}+ √(v_{i}^{2}+ 2gy)]/g

Since **y** is below the starting point, it also is a positive number.

Time as a function of downward velocity or displacement

## Examples

The following examples illustrate applications of the equations.

### Time for a given velocity

If you throw a ball downward from a tall building at 5 ft/s, find the time it takes for the ball to reach a velocity of 101 ft/s.

#### Solution

You are given that **v _{i}** = +5 ft/s and

**v**= 101 ft/s. Since

**v**and

_{i}**v**are in ft/s, then

**g**= 32 ft/s

^{2}. The equation to use is:

t = (v − v_{i})/g

Substitute values in the equation:

t =(101 ft/s − 5 ft/s)/(32 ft/s^{2})

t= (96 ft/s)/(32 ft/s^{2})

t= 3 s

### Time for a given displacement

If you throw an object downward from a high building at 5 m/s, find the time it takes to fall 50 m.

#### Solution

You are given that **v _{i}** = +5 m/s and

**y**= 50 m. Since

**v**in m/s and

_{i}**y**is in m, then

**g**= 9.8 m/s

^{2}. The equation to use is:

t = [−v_{i}+ √(v_{i}^{2 }+ 2gy)]/g

Substitute values in the equation:

t = [−5 m/s +√{(25 m/s)² + 2*(9.8 m/s²)*(50 m)}]/(9.8 m/s²)

t = [−5 m/s +√(625 m²/s² + 980 m²/s²)]/(9.8 m/s²)

t = [−5 m/s +√(1605 m²/s²)]/(9.8 m/s²)

t = [−5 m/s + 40.1 m/s]/(9.8 m/s²)

t =(35.1 m/s)/(9.8 m/s²)

t =3.58 s

(Whew!)

## Summary

You can calculate the time it takes an object that is projected downward to reach a given velocity or reach a given displacement from the starting point from the equations:

t = (v − v_{i})/g

t = [−v_{i}+ √(v_{i}^{2 }+ 2gy)]/g

Gravitate toward good grades

## Resources and references

### Websites

**Equations for a falling body** - Wikipedia

**Gravity Calculations - Earth** - Calculator

**Kinematic Equations and Free Fall** - Physics Classroom

### Books

**Top-rated books on Simple Gravity Science**

**Top-rated books on Advanced Gravity Physics**

## Questions and comments

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## Gravity Time Equations for Objects Projected Downward