What happens when an object is thrown downwards?

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Velocity with respect to time
For a given displacement
For a given time
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Physicists would say that the ball has been given an initial velocity. But this initial velocity doesn’t affect the force of gravity on the ball, so the ball would still gain a downward velocity of 9.8 m/s each second it is in flight.

Suppose the ball is thrown down with a velocity of 2.0 m/s. Then at time t = 0, the time it was thrown down, it would already have a velocity of 2.0 m/s downward. If you use the table above you see that 0.10 s later it would have a velocity of 2 0.98 m/s = 2.98 m/s. Half a second after launch its downward velocity would be 2 4.9 m/s = 6.9 m/s. That is, the initial velocity is just added to the increasing velocity caused by the gravitational force.

Now suppose the ball is thrown upward with the same 2.0 m/s speed. Now we have to be very careful with the sign of the velocity. We have chosen the downward direction as positive. So the initial velocity of the ball is actually –2.0 m/s.

The gravitational force acts in the downward direction. By Newton’s Second Law that means that the ball will slow down at a rate of 9.8 m/s each second. After 0.1 s it would be going -2.0 0.98 = -1.02 m/s. After 0.20 s it would be going -2.0 1.96 = -0.04 m/s. It is almost at rest! But the gravitational force is still acting on it, so it continues to accelerate downward. After 0.30 s it would be going -2.0 2.94 = 0.94 m/s. That is, it would now be moving downward.

When it is at its maximum height does it stop? At an instant in time it is indeed at rest, but it doesn’t stay motionless for any time interval because the gravitational force keeps acting on it.

SfC Home > Physics > Gravity >

by Ron Kurtus

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 velocity when the object reaches a given displacement from the starting point or when it reaches a given elapsed 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 velocity for a given displacement?
How do you find the velocity for a given time?
What are some examples of these equations?

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

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

v = ±√(2gy + vi2)

where

± means plus or minus
v is the vertical velocity in meters/second (m/s) or feet/second (ft/s)
√(2gy + vi2) is the square root of the quantity (2gy + vi2)
y is the vertical displacement in m or ft
g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)
vi is the initial vertical velocity of the object

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

Since v is a downward vector, it has a positive value. Likewise, y and vi are positive numbers. Thus, only the + version of the equation applies:

v = √(2gy + vi2)

Velocity of object projected downward as a function of displacement or time

Velocity with respect to time

The general gravity equation for velocity with respect to time is:

v = gt + vi

where t is the time the object has fallen in seconds (s).

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

This same equation applies for an object projected downward.

Examples

The following examples illustrate applications of the equations.

For a given displacement

Find the velocity of a rock that is thrown down at 2 m/s after it has traveled 2 meters.

Solution

You are given that vi = 2 m/s and y = 2 m. Since vi is in m/s and y is in meters, then g = 9.8 m/s2. The equation to use is:

v = √(2gy + vi2)

Substitute values in the equation:

v = √[2*(9.8 m/s2)*(2 m) + (2 m/s)2]

v = √(39.2 m2/s2 + 4 m2/s2)

v = √(43.2 m2/s2)

v = 6.57 m/s

For a given time

Suppose you throw the object downward at 10 m/s. Find its velocity after 4 seconds.

Solution

You are given that vi = 10 m/s and t = 4 s. Since vi is in m/s, g = 9.8 m/s2. The equation to use is:

v = gt + vi

Substitute values in the equation:

v = (9.8 m/s2)*(4 s) + 10 m/s

v = 39.2 m/s + 10 m/s

v = 49.2 m/s

Summary

You can calculate the velocity when an object that is projected downward reaches a given displacement from the starting point or when it reaches a given elapsed time from the equations:

v = √(2gy + vi2)

v = gt + vi

Help other people learn

Resources and references

Ron Kurtus' Credentials

Websites

Gravity Resources

Equations for a falling body - Wikipedia

Gravity Calculations - Earth - Calculator

Kinematic Equations and Free Fall - Physics Classroom

Books

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Top-rated books on Simple Gravity Science

Top-rated books on Advanced Gravity Physics

Questions and comments

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