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A police officer is standing near a highway using a radar gun to catch speeders. He aims the gun at car that has just passed his position and, when the gun is pointing at an angle of 45 to the direction of the highway, notes that the distance between the car and the gun is increasing at a rate of 100 km/h. How fast is the car travelling?

Task

I have done this task using simple geometry:

$$x^2+k^2=s^2$$ $$x *dx/dt=s*ds/dt$$ Then since $$x=k$$ and $$s=\sqrt{2}k$$ I find out the speed is about 141 km/h.

However, I struggle with task b of this problem:

b) If the radar gun in a is aimed at a car travelling at 90 km/h along a straigt road, what will its reading be when it is aimed making an angle of 30 degrees with the road?

So the car is travelling at 90 km/h says the text, but won't the radar be reading 90 km/h then? Or are those 90 km/h equal to ds/dt, because then I will be able to solve the task.

Mathomat55
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1 Answers1

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The radar gun can only measure the rate at which the distance between the gun and the car is increasing. In your notation, the radar gun measures $ds/dt$. The speed of the car, on the other hand, is represented by $dx/dt$ in your notation.

So to setup your solution to part b, you'll enforce that $dx/dt = 90$ km/h. Also note that if the radar gun makes an angle of 30 degrees with the road, the relationship $x = k$ will no longer hold. Rather, you will find $k/x = \tan{30^{\circ}}$ and $k/s = \sin{30^{\circ}}$. With this information you can return to your rate relation and determine $ds/dt$, the reading on the radar gun, in terms of $k$.

JoeBingo
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