The problems on this worksheet were adapted from this Math Overflow post. The
first problem is from Robert Israel, and the second is from the user KConrad.
1. Doppler radar measures the rate of change of the distance from an object to
the observer.
(a) A police officer a meters away from a straight road points a radar gun at
a car traveling along the road, c meters away, and measures a speed of v.
What is the car’s actual speed?
(b) Suppose that in the setup of the previous problem, the officer is 10 meters
away from the road, the car is 100 meters down the road, the radar
gun measures a speed of 50mph, and the speed limit is 50mph. Is the
driver over the speed limit? (Hint: You shouldn’t have to do any explicit
computations.)
(c) How should a police officer trying to avoid doing any arithmetic set up
their speed trap?
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2. Suppose that a ladder leaning against a wall starts slipping down. Assume
the part of the ladder touching the ground is moving away from the wall at a
constant rate.
(a) Before doing any math, predict whether the part of the ladder touching
the wall will fall at a constant rate. (And try to avoid looking at the later
parts of the question because they’ll give the answer away.)
(b) Now use calculus to check your answer in part (a).
(c) In the moments just before the ladder finishes falling, how quickly does
your model predict the part of the ladder touching the wall will be moving? Does this make physical sense? Do you still believe your answer to
part (b)?
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