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How to Measure the Speed of a Car around a Race Track

One of the great things about trigonometry is that you can use it to measure faraway things — or things that you don’t want to get too close to, like a race track.

A car races around the track.
A car races around the track.

Let’s say a race car is going around a circular track. A photographer standing at the center of the track takes a picture, turns 80 degrees, and then takes another picture 10 seconds later. If the track has a diameter of 1/2 mile, how fast is the race car going? The preceding figure shows the photographer in the middle and the car in the two different positions.

How fast is the car going? Where does the problem make any mention of speed? Actually, in this situation, the car travels partway around the track in 10 seconds. If you can find out how far the car goes in that time, then you can figure out the car’s speed by using the distance formula, where distance is equal to rate multiplied by time.

A formula that you’ll find mighty helpful is: d = rt, which says distance equals rate multiplied by time. If the rate is miles per hour (or feet per second or some such measure), then time has to be the same measure as in the rate.

Now, to find out how fast the racecar is traveling:

  1. First, change the 80 degrees to radians.

  2. Input the numbers in the arc-length formula.

    Putting in the radian measure and the radius of the track (1/4 mile), you get


    which is the distance the car traveled in 10 seconds.

  3. Multiply this result by 6 to get miles traveled in a minute.

    This calculation gives you 2.094 miles per minute.

  4. Then multiply that number by 60 to get miles traveled in one hour.

    This calculation gives you 125.64 miles. So the car is traveling about 125 mph.

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