Falling objects will form an interesting class of motion problems. For example, you are able to estimate the depth of a vertical mine shaft by dropping a rock into it, and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, you are able to examine several interesting situations and learn much about gravity in the process. Well, here we are going to talk about gravity velocity equations for falling objects.

**Gravity Velocity Equations for Falling Objects**

When you drop an object from several heights above the ground, it has an initial velocity of zero. Simple equations will allow you to calculate the velocity a falling object reaches after a given period of time, and its velocity at a given displacement. The equations will assume that air resistance is negligible.

**Velocity with respect to time**

In general, the gravity equation for velocity with respect to time is:

v = gt + vi

Since the initial velocity vi = 0 for an object which is falling, so the equation reduces to:

v = gt

where,

- v is the vertical velocity of the object in meters/second or feet/second
- g is the acceleration due to the gravity (9.8 m/s2 or 32 ft/s2)
- t is the time in seconds that the object has fallen

**Velocity with respect to displacement**

In general, the gravity equation for velocity with respect to displacement is:

v = ±√(2gy + vi2)

where

- ± has meaning plus or minus
- √(2gy + vi2) is the square root of quantity (2gy + vi2)
- y is vertical displacement in meters/feet

Since vi = 0, y is positive, as it is below the starting point. Also, v is downward and positive. Only the + term of ± applies.

So, the equation for the velocity of a falling object after it has traveled a certain displacement is: v = √(2gy)

**Examples**

The examples below illustrate the applications of the equations.

**For a given time**

What will be the velocity of an object after it falls for three seconds?

**Solution**

Substitute in the equation below:

v = gt

If you use g = 9.8 m/s2, then, v = (9.8 m/s2)*(3 s) = 29.4 m/s.

If you use g = 32 ft/s2, then, v = (32 ft/s2)*(3 s) = 96 ft/s.

**For a given displacement**

What is the gravity velocity of an object after it has fallen one hundred feet?

**Solution**

Because y is in feet, then g = 32 ft/s2. Substitute in the equation below:

v = √(2gy)

v = √[2*(32 ft/s2)*(100 ft)]

v = √(6400 ft2/s2)

v = 80 ft/s

**Formula for the Speed of a Falling Object**

As an object falls, its speed will increase because it’s being pulled on by gravity. The acceleration of gravity near the earth is g = -9.81 m/s^2. To discover something’s speed (or velocity) after a certain amount of time, you have to multiply the acceleration of gravity by the amount of time since it was let go of. So, you are going to get: velocity = -9.81 m/s^2 * time, or V = gt. The negative sign only means that the object is moving downwards. If it were positive, then it would be moving up. For speed rather than velocity, you only drop the negative sign.

If you have an initial velocity (if you threw the ball up or down), then you must include this in the equation, giving you: V = Vo + gt, where Vo is the initial velocity of the object. This equation still works if you throw the ball to the side, instead of straight up or down, except that it will give you the up-down velocity, not the total velocity.

**How to Calculate Velocity of Falling Objects?**

Two objects of different mass dropped from a building will strike the ground simultaneously. This occurs as the acceleration due to gravity is constant at 9.81 meters per second per second (9.81 m/s^2) or 32 feet per second per second (32 ft/s^2), regardless of mass. As a result, gravity is going to accelerate a falling object so its velocity increases 9.81 m/s or 32 ft/s for every second it experiences free fall. Velocity can be calculated via v = gt, where g represents the acceleration due to gravity and t represents time in free fall. Then, the distance traveled by a falling object is calculated via d = 0.5gt^2. Also, the velocity of a falling object can be determined from time in free fall or from distance fallen.

**Known Time**

You have to convert all units of time to seconds. For instance, an object that falls for 850 milliseconds falls for 0.850 seconds.

Calculate the metric solution of velocity by multiplying the time in free fall by 9.81 m/s^2. For an object which falls for 0.850 seconds, the v = 9.81 m/s^2 * 0.850 s = 8.34 m/s.

You are able to determine the imperial solution by multiplying the time in free fall by 32 ft/s^2. Continuing the example before, v = 32 ft/s^2 * 0.850 = 27.2 ft/s. For a result, the velocity of the falling object in the example is 27.2 feet per second.

**Known Distance**

You have to convert all units of distance fallen to units of feet or meters by using an online unit conversion tool. A distance of 88 inches, for instance, represents 7.3 feet or 2.2 meters.

Calculate the time during free fall according to t = [d / (0.5g)]^0.5 that represents the equation d = 0.5gt^2 solved for time. For an object which falls 2.2 meters, t = [2.2 / (0.5 * 9.81)]^0.5, or t = 0.67 seconds. In alternative, t = [7.3 / (0.5 * 32)]^0.5 = 0.68 seconds.

You are able to determine the velocity at the moment of impact according to v = gt. Continuing the example before, v = 9.81 * 0.67 = 6.6 m/s or v = 32 * 0.68 = 21.8 ft/s. As a result, the velocity of the falling object in the example is 21.8 feet per second.