# Vertical Velocity: Definition & Equation

## Definition of Vertical Velocity

You go skydiving, and after hurling yourself from an airplane, you deploy the parachute. You throw a ball upward and catch it as it falls back down. These are examples of situations that involve vertical velocity.

First, let's define velocity. **Velocity** is a mathematical quantity that tells us how fast your position is changing. For instance, if you move 30 meters in 10 seconds, then your velocity is 30 meters divided by 10 seconds, or 3 meters a second. Velocity is a vector quantity, so we must specify the **magnitude** of the velocity, which indicates the size, and the **direction** of the velocity, which indicates where it is going. In one-dimensional motion, the magnitude is just how big the rate of change is, for example, 30 meters a second. The direction is denoted by a plus (+) sign for up and right directions, and a minus (-) sign for down and left directions.

**Vertical velocity** is a special type of velocity because in the vertical direction, it is always affected by **acceleration due to gravity**. Any object thrown up, thrown down, or dropped in the vertical direction is affected by this acceleration, which has a magnitude of about 10 meters/second/second, or 10 meters/second squared, directed downward, toward the center of the earth. The saying 'what goes up must come down' is a perfect description of vertical velocity. The gravity of the earth will cause objects to fall back down to the earth at a rate of about 10 meters/second/second.

## Calculating Vertical Velocity

Take a look at this formula for calculating vertical velocity.

Calculating vertical velocity is a bit different than calculating general and horizontal velocity, because of the acceleration due to gravity, which is denoted by the letter 'g' in the equations. Whenever you see 'g' in an equation, it refers to 10 meters/second/second. In order to calculate the vertical velocity of an object at any time 't', we need to know the initial velocity and the time of interest.

Since velocity is a vector, the vertical velocities you calculate using this equation can be positive, negative, or zero. Let's look at some examples.

A ball is dropped from a tall building. What is the vertical velocity of the ball after 5 seconds? Note that the term 'dropped' always means that the initial velocity is zero.

Note that the velocity is negative because the object is moving downward, which is our negative direction.

If a boy uses a device to launch a rock straight up, with an initial velocity of 35 meters a second, what is the vertical velocity of the rock after 3 seconds? Any object thrown up has a positive velocity, and any object thrown down has a negative velocity. Note that the velocity is positive and smaller than the initial velocity. This means that the rock is moving up and that gravity slowed down the velocity of the rock by a significant amount. The rock will eventually stop, and drop back down to the ground.

## Lesson Summary

**Vertical velocity** is a special case of velocity only because the acceleration due to gravity acts in the vertical direction. This means that any motion in the vertical direction will be affected by the acceleration due to gravity. How? Gravity acts to bring down objects launched in the vertical direction. You have to be going fast to escape the gravity of the earth (usually referred to as escape velocity). Any velocity below this velocity will end up falling back to the earth.

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## Understanding the Acceleration of Vertical Velocity

If an object is launched vertically away from the Earth, it will slow down as it moves upward. Eventually, it will come to a complete stop. The object will only be at rest momentarily as it then begins to fall in the opposite direction back toward the center of the Earth. But what about the moment when it comes to rest at the top of its motion; how can we say that it is accelerating if for that moment it is not moving? As a reminder, the definition of acceleration, {eq}\mathbf{a}
{/eq}, is a change in velocity, {eq}\mathbf{v}
{/eq} over time, *t*, or

{eq}\mathbf{a} = \Delta \mathbf{v} / \Delta t {/eq}.

#### Q: How can an object at the pinnacle of its vertical motion be accelerating if its vertical velocity is zero?

The answer can be seen in two ways:

a. The cause of acceleration is a net force. As long as the net force is acting on an object it will accelerate, or an object will only accelerate because there's a net force on it. What is the cause of acceleration for an object displaying Vertical Velocity? The answer is gravity pulling it down toward the center of the Earth. Inside we said that objects that are displaying vertical are in free fall which means ignoring air resistance the only force on the object is gravity pulling downward. So the only force on an object is gravity pulling downward then gravity is a net force on the object during its entire motion. As long as there's a net force on the object there will be acceleration. Acceleration is always in the direction of the net force. So gravity, being the net force, is constantly pulling downward then the acceleration will be constant and always be downward, even when is momentarily at rest.

b. A second way to understand how an object can accelerate when it is at rest even though it's not moving, is to question what would happen to it if it didn't accelerate? Remember that acceleration is a change in velocity. When the object is momentarily at rest if it does not change its velocity, what will it continue to do? If an object is at rest and does not change, it will stay at rest. Does the object in Vertical Velocity stay at rest at the top of its motion? The answer is no; it continues to change its motion and begins to fall back toward the Earth.

So as long as a net force is acting on the object the object will continue to accelerate in the direction of the net force.

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