**Contents**:

- What is Velocity?
- The Velocity Function
- Finding velocity with calculus
- Average Velocity
- Velocity of a Falling Object:

**See also**: Instantaneous Velocity

## What is Velocity?

In every day use,*velocity*is often used interchangeably with

*speed*. However, in calculus there is a significant difference between the two. In mathematics,

*speed*refers to how fast an object travels.

*Velocity*, however, is defined as both the rate at which an object is traveling (i.e. how fast) and its

**direction**. In calculus, it is the first derivative of the position function.

As **calculus** is the mathematical study of rates of change, and velocity is the measure of the change in position of an object with respect to time, the two come in contact often. Finding the velocity of simple functions can be done without the use of calculus. However, when the graph of a function is curved the change isn’t constant, and performing calculus operations becomes necessary.

## The Velocity Function

This gives you an object’s rate of change of position with respect to a reference frame (for example, an origin or starting point), and is a function of time. It tells the **speed **of an object *and *the **direction **(e.g. 70 km/h south). It is usually denoted as v(t). For example, v(t) = 2x^{2} + 9.

Note that this is different from the *velocity equation* you probably came across in algebra. The *equation ***v = S/T** gives you the average velocity of an object, given distance and time. The *function *enables you to find **instantaneous change** as well as average change. It must also meet the requirements for being a function.

## Sign

The **sign of the function** tells you the direction the object is traveling.

**Positive**: v(t) > 0 = motion to the right on the number line (i.e. in a positive direction).**Negative**: v(t) < 0 = motion to the left on the number line (i.e. in a negative direction).

If an object is traveling in a straight line (i.e. with rectiliear motion), you can use the sign to tell you whether the object is speeding up or slowing down:

**Slowing down**: velocity and acceleration have opposite signs (e.g. + -).**Speeding up**: they have the same sign (e.g. + +).

Watch the video for an **example of to use the function to find total distance traveled.**

## How to Get the Find The Velocity Function

You can use differentiation and integration to find the function. The following image shows the relationship between the first few derivatives and integrals as they relate to the velocity function:

The derivative of the position function gives you the velocity of a moving object, assuming the object is a/ moving in a straight line and b/ air resistance is zero. More formally, we say that the velocity of an object is the rate of change of an object’s position, with respect to time.

As an example, let’s say you were given a position function. You can differentiate it (i.e. take the first derivative) to get the velocity function. For an example of how to do this, see: Velocity of a Falling Object.

On the other hand, if you have a jerk function you’re going to want to work backwards. In other words, you need to integrate the function. How you do this depends on what type of function you have. See: Integral Rules.

## Finding Velocity

How you solve for a falling object depends upon what information you’ve been given. For example, if you’ve been given a time (usually in seconds), then the velocity of any falling object can be found with the equation v = g * t, where g is acceleration due to gravity. However, if you’ve been given a position function (e.g. for the height), then you need a little calculus to derive the answer.

If you are given the position function (for the distance moved by the object), you can take the **derivative** and reach the function. If you are given the formula for the object’s acceleration, you can find the **integral** and again come to the solution.

## 1. Given a Position (Displacement) Function

**Example question: **Find the velocity for the following position function:x(t) = 4t^{2} + 4t + 4

Step 1: In order to get from the displacement to the velocity, you will take the **derivative** of the displacement with respect to time. First, set up your equation to get ready to find a derivative:

x(t) = 4t^{2} + 4t + 4

dx/dt = d/dt 4t^{2} + 4t + 4 = v(t)

Step 2: Solve for the derivative. Here you can use the Power Rule, and rule for derivative of constants.

d/dt 4t^{2} + 4t + 4 = 8t + 4

v(t) = 8t + 4

*That’s it!*

Tip: The velocity is not constant over time, so *t* makes an appearance. If it were constant, it would not have the variable in it, and it would also have an acceleration of 0.

## 2. Find velocity given Acceleration.

**Example question: **Find the velocity from the following acceleration function:

a(t) = 10t + 5

Step 1: Set up the equation to perform an integration:

a(t) = 10t + 5

v(t) = ∫ a(t) dt = ∫ 10t + 5 dt

Step 2: Perform **integration** on a(t). Integration is a somewhat advanced calculus method, so be sure to take a look at the articles specifically detailing it (see: Integrals) if you are unfamiliar with it.

∫ a(t) dt = ∫ 10t + 5 dt = 5t^{2} + 5t + c = v(t)

*Note:* c is a constant of integration that can’t be determined without more information. If you are supplied an initial value, you can find the constant by setting time equal to 0. Using the above for example:

v(0)= 5(0^{2}) + 5(0) + c = c = v(0)

## Average Velocity

Average velocity is defined in terms of the relationship between the distance traveled and the time that it takes to travel that distance. The formula for finding average velocity is:

v_{av} = x_{f} – x_{i }/ t_{f} – t_{i}

Where:

- v
_{av }is the average velocity, - x
_{f }is the final position of the object, - x
_{i }is the initial position of the object, - t
_{f}is the final time, - t
_{i}is the initial time.

## Example:

A car starts at position x = 16 feet. After 8 seconds the car is 134 feet east of its initial position. What is the car’s average velocity?

Step One:**Find the difference between the initial and final positions **of the car.

The car traveled from 16 feet to 134 feet (134 – 16 = 118). The car traveled east a total of 118 feet.

Step Two:**Calculate the amount of elapsed time,** if it isn’t explicitly stated. We know that the car traveled for 8 seconds. No further calculations are required here.

Step Three:**Plug the values into the formula and solve**.

V_{av} = 118ft / 8s = 14.75 ft/s east

The average velocity that the car traveled was 14.75 feet-per-second eastward.

## 1. Velocity of a Falling Object: v = g*t

A falling object is acted on by the force of gravity: -9.81 m/s^{2} (32 ft/s). Gravity will accelerate a falling object, increasing its velocity by 9.81 m/s (or or 32 ft/s) for every second it experiences free fall.

In order to find the velocity of a particular falling object, just multiply time (t) by gravity (t). The formula is:

**v = g*t
v = -9.81 m/s ^{2}*t**

**Example #1**: An object falls for 1.2 seconds. What is it’s velocity?

- v = -9.81 m/s
^{2}*t - v = -9.81 m/s
^{2}*1.2s **v = 11.77200 m / s**^{2}

## 2. Velocity of a Falling Object Using Calculus

Calculus is very useful for finding the velocity of a falling object if all you have is a position function, like the height of an object. First, differentiate the position function to get the velocity function. Then use *that *function to find the answer.

## Example #2: Position Function

The function v(t) is the derivative of the position function. If you’re given a position equation like h(t) or s(t), you’ll need to differentiate that function in order to find the velocity of the falling object.

**Example problem:** Frustrated with your calculus class, you throw your textbook out of your dorm window, which is 200 feet above your car in the parking lot. The height of the book, in feet over the car after t seconds is given by the function h(t) = 200 – 16t^{2}. The book will dent your car if it’s going more than 100 feet per second. Will your car get dented?

**Hint:**The given equation is *not *for the velocity of a falling object. It is a position function.

Step 1: **Differentiate the position function,** h(t) = 200 – 16t^{2} to get the function (you need to know the velocity to answer the question).

- 200 is a constant, so it disappears.
- 16t
^{2}can be differentiated using the power rule.

The differentiated function is 2(16)t^{2-1} = -32t.

Step 2: **Solve the position function for zero ** (in other words, when the height is zero) to find out when the book will hit the car. You know the function from Step 1.

Setting h(t) = 0 gives:

- 0 = 200 – 16t
^{2} - t
^{2}= 200/16 = 12.5 - t = 3.54

Step 3: **Insert your answer** from Step 2 into the function from Step 1:

- v(5) = -32(5)
- v(5) = -113.28

The velocity is 113.28 feet per second when the book hits the car, which is more than 100 feet per second. Yes, there will be a dent!

*That’s it!*

**Tip**: A negative sign indicates the height is decreasing.

## Other Useful Equations

If you are given the height/distance the object has fallen (d), then use this equation to find the time (t):

If you are given the height/distance the object has fallen (d), then use this equation to find the **instantaneous velocity** of the object:

If you are given the height/distance the object has fallen (d), then use this equation to find the **average velocity** of the object:

## References

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