How to Calculate Linear Velocity

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Have you ever wondered how scientists are able to figure out the Earth's speed as it travels around the Sun? They don't do it by measuring the time it takes for the planet to pass a pair of reference points, because there are no such references in space. They actually derive Earth's linear velocity from its angular velocity using a simple formula that works for any body or point in circular rotation around a central point or axis.

Period and Frequency

When an object is rotating around a central point, the time it takes to complete a single revolution is known as the ​period​ (​p​) of rotation. On the other hand, the number of revolutions it makes in a given period of time, usually a second, is the the ​frequency​ (​f​). These are inverse quantities. In other words:

p=\frac{1}{f}

Angular Velocity Formula

When an object travels on a circular path from point ​A​ to point ​B​, a line from the object to the center of the circle traces an arc on the circle while sweeping out an angle at the center of the circle. If you denote the length of the arc ​AB​ with the letter "​s​" and the distance from the object to the center of the circle "​r​," the value of the angle (​ø​) swept out as the object travels from ​A​ to ​B​ is given by

\phi= \frac{s}{r}

In general, you calculate the average angular velocity of the rotating object (​w​) by measuring the time (​t​) it takes for the the radius line to sweep out any angle ​ø​ and using the following formula:

w= \frac{\phi}{t} \; (\text{rad/s})

ø​ is measured in radians. One radian is equal to the angle swept when the arc ​s​ is equal to the radius ​r​. It's about 57.3 degrees.

When an object makes a complete revolution around a circle, the radius line sweeps out an angle of 2π radians, or 360 degrees. You can use this information to convert rpm to angular velocity, and vice versa. All you need to do is measure the frequency in revolutions per minute. Alternatively, you can measure the period, which is the time (in minutes) for one revolution. The angular velocity then becomes:

w = 2πf = \frac{2π}{p}

Linear Velocity Formula

If you consider a series of points along a radius line moving with an angular velocity of ​w​, each one has a different linear velocity (​v​) depending on its distance r from the center of rotation. As ​r​ gets larger, so does ​v​. The relationship is

v=wr

Since radians are dimensionless units, this expression gives the linear velocity in units of distance over time, as you would expect. If you have measured the frequency of rotation, you can directly calculate the linear velocity of the rotating point. It is:

v = (2πf)×r
v = \bigg(\frac{2π}{p}\bigg)×r

How Fast Is the Earth Moving?

To calculate the velocity of the earth in miles per hour, you need only two pieces of information. One of them is the radius of the Earth's orbit. According to NASA, it's 1.496 × 108 kilometers, or 93 million miles. The other fact you need is the period of the Earth's rotation, which is easy to figure out. It's one year, which is equal to 8760 hours.

Plugging these numbers into the expression ​v​ = (2π/​p​) × r tells you that the linear speed of the earth traveling around the sun is:

\begin{aligned} v &= \bigg(\frac{2×3.14}{8760 \; \text{hours}}\bigg)×9.3 ×10^7 \;\text{miles} \\ &=66,671 \text{miles per hour} \end{aligned}

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