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?
Solution. First let’s pick a coordinate system. We’ll put the officer at
the origin, and let the road be parallel to the x-axis. Let r(t) be the
distance from the car to the officer at time t, and let x(t) be how far
down the road the car is at time t. Then the Pythagorean theorem tells
us x2 + a2 = r2 . Now we implicitly differentiate to get
2x
dx
dr
= 2r ,
dt
dt
so
r dr
dx
=
.
dt
x dt
When the officer takes the measurement, x = c, r2 = c2 +a2 , and dr/dt =
v. Therefore,
r
a 2
dx
= 1+
v.
dt
c
(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.)
q
2
Solution. Since 1 + ac
> 1, we can see that the radar gun will
always underestimate the speed of the car. Therefore, the driver is over
the speed limit.
(c) How should a police officer trying to avoid doing any arithmetic set up
their speed trap?
q
2
Solution. As c gets large, 1 + ac ≈ 1. Therefore, if the officer takes
measurements when the car is very far down the road, the speed reported
by the radar gun will be close to the driver’s actual speed. So the officer
should pick a long stretch of straight road and try to take measurements
when the car is as far away as possible.
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2. Suppose that a ladder of 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).
Solution. Suppose the length of the ladder is L. Let x(t) be the position
of the part of the ladder touching the ground at time t, and let y(t) be
the position of the part of the ladder touching the wall at time t. Then
x2 + y 2 = L2 , and implicit differentiation gives us
2x
dx
dy
+ 2y
= 0,
dt
dt
Now solving for dy/dt and using the relation y =
√
L2 − x2 gives us
dx
dy
x
=√
.
2
2
dt
L − x dt
√
Now dx/dt at t varies, but x/ L2 − x2 certainly isn’t constant, so dy/dt
isn’t constant either.
(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)?
Solution. The model predicts that dy/dt → ∞ as x → L. This doesn’t
make physical sense. I’d still believe my answer to part (b) for two
reasons. First, the result is qualitative rather than quantitative. Second,
I’d guess the model stops working when x gets close to L because the
ladder pulls away from the wall. Our physical intuition says that the
ladder will be on the wall for some period, and I’d expect our model to
be roughly accurate during that period.
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