# Linear Speed

Hi, and welcome to this video about linear speed!

Imagine running forward and jumping onto a spinning merry-go-round. At first, you are running in a straight line, but as you land, your motion changes and you begin to spin. How is your running speed related to your spinning speed? Let’s find out!

First, recall that speed is the distance traveled over a particular time interval. Cars have a speedometer, which tells us how fast the car is going in miles per hour– miles as the units of distance and hours as the units of time. Linear speed tells us how fast a point on a rotating object is traveling. The distance the point travels depends on how far it is from the center of rotation. A point further from the center of rotation travels farther than a point closer to the center of rotation. The time is how long it takes for the object to complete one revolution.

We calculate the linear speed of a point using a few different methods. First, we need to find the distance traveled by the point, and to do that, we need to know how far the point is from the center of rotation; this is the radius of the circle the object creates while rotating. Let’s call this r. Next, we need to measure how far the point has rotated. This is the angle the object has traveled through, θ, which is measured in radians or degrees. Once we have these two measurements, we calculate the arc length, which is the distance the point has traveled. We call this distance s, and the equation we use is s=rθ. Now that we have the distance traveled, we measure the time taken to travel this distance, t, and find the linear speed: v = s/t = rθ/t.

Sometimes we know how fast something is rotating and we want to calculate the linear speed. Usually, we know how many revolutions per minute the object makes, so let’s connect RPM and linear speed.

Recall an angle can be measured with two different units: radians or degrees. A full revolution is 2π radians or 360 degrees. When converting between rotating speed and linear speed, the units we use depend on the final answer we want. Let’s use radians first.

Suppose we have a wheel spinning at ω RPM with a radius of r meters. What’s the linear speed in meters per second of a point on the outermost edge of the tire?

We can use simple unit analysis to find our answer! First, we know our wheel spins through 2π radians each revolution. But how do we relate radians to a length with units of meters? Imagine wrapping our wheel with a piece of string. This string makes one revolution around our wheel. It “measures” 2π radians. Now, let’s take that piece of string, lay it flat, and measure its length with a ruler. We will find the length is 2πr meters. So with each revolution, our wheel travels 2πr meters. To convert from minutes to seconds, recall there are 60 seconds in one minute. So we see,

In general, the formula is v = 2πrω, where r is the distance from the rotating point to the center of rotation and ω is how fast the object is spinning. Let’s look at an example problem:

The earth travels one revolution around the sun in one year. It is approximately 93 million miles from the sun. What is the earth’s linear speed in miles per hour? Let’s use the formula v=2rwith some unit changes. We have ω in units of revolutions per year. We can follow unit analysis to find the linear speed. One year has 365.25 days (there is a leap year every four years, which is equivalent to adding one quarter of a day to each year), and one day has 24 hours. Then, one rotation is 2πr, where r is 93 million miles. So we see the earth is zipping around at 66 THOUSAND miles per hour!

But what if we are using degrees to measure an angle instead of radians? Our wheel with radius r is now rotating at 60 degrees per minute and we would like to know the linear speed in feet per second. The easiest way is to add in one more unit conversion: convert from degrees to rotations! Then you can use the same steps to find the linear speed!

In general, the formula is v = 2πrω/180, where r is the distance from the rotating object to the center of rotation and ω is how fast the object is spinning in degrees per second.

Let’s look at another example:

On Chicago’s Navy Pier, the Centennial Ferris Wheel lifts passengers nearly 200 feet high. One ride lasts approximately 15 minutes and you get to go around 3 times. What is your average linear speed in feet per second?

First, we are given the diameter of the Ferris wheel, not the radius. Recall the diameter of a circle is 2 times the radius, so r =100 feet. Next, we need to figure out how fast we are rotating. We rotate 3 times in 15 minutes, which is equal to 1 rotation per 5 minutes. Now we can use

v = 2πrω to find our linear speed to be approximately 2 feet per second.

Thanks for watching, and happy studying!