Question

A highway patrol officer is seated on a motorcycle on a curvy section of Highway 1. The posted speed limit is 45 miles per hour on this stretch of highway. The officer is monitoring traffic using radar. The next exit is 3.6 miles up the road. The radar picks up a speeding car averaging 68 mph. When the officer tries to start his motorcycle to follow the car, it won’t start. He tries again and again, and soon he fears that he won’t be able to catch the speeding car before it turns off the highway. Finally, his motorcycle starts and he begins the pursuit 30 seconds after the speeding car has passed him on the roadside.

How fast does the officer need to go to catch up to the speeding car? What is his average speed in pursuit?

Answer 1

80.7 mph

Explanation:

In order to catch the car at the exit, the officer must cover the distance in 30 seconds less time than the car does. The time, speed, distance relation can be used.

Time for car to reach the exit

The relation between time, speed, and distance is ...

time = distance/speed

Here, speed is in miles per hour, and distance is in miles. That means time will be given in hours by the straightforward application of this formula.

time = (3.6 mi)/(68 mi/h) = (3.6/68) h = 9/170 h

Time for the motorcycle to reach the exit

There are 3600 seconds in 1 hour, so 30 seconds represents this fraction of 1 hour:

30/3600 = 1/120

The motorcycle will have this amount of time less than the time the car takes to reach the exit, so ...

(9/170 -1/120) h = (9(120)-170)/(170×120) h = 910/20400 h = 91/2040 h

Motorcycle speed

The speed required to cover 3.6 miles in 91/2040 hours is ...

speed = distance/time

(3.6 mi)/(91/2040 h) = 3.6×2040/91 mph = 7344/91 mph ≈ 80.7 mph

The officer's average speed in pursuit must be 80.7 miles per hour (or faster).

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Additional comment

There are other ways the problem can be worked. One of them is to consider the remaining distance the car must travel when the officer begins pursuit. That will be 3.6 mi less (1/120 h)(68 mi/h) = 17/30 miles. The ratio of the officer's speed to the car's speed will be the same as the ratio of the officer's miles to the car's miles: 3.6/(3.6 -17/30) × 68 mph ≈ 80.7 mph.

Intermediate values used in the calculations should be kept at full calculator precision. Rounding should only be done at the end. Here, the actual speed required is 80.7032967... mph, rounded down for the above result.