Relativity Problem
You are a swimmer in training for the Olympics. One day you and a teammate go out
to Martha Lake to train. When your coach times you it is discovered that both you
and your friend swim at exactly the same speed: c meters/sec.
A couple of days later you and your friend go out to the Snohomish River to do a
little training. Upon arriving a heated argument breaks out between your coach and
your friend as to which is harder, doing laps that go across the river or doing laps
that go upstream and then downstream. You decide that the best way to settle the
argument is to do the experiment.
You decide to hold a race with one of the swimmers going upstream a distance equal
to the distance across the river and then swimming back downstream to the
starting point. The other swimmer is to swim across the river and back.
Assume that both swimmers have the same speed in still water and also that the
current in the river is a constant speed of v' at every point in the river.
Assume that the time for the swimmer going across the river is t and the time for
the swimmer going up/down stream is t’.
Assume the distance across the river is distance d.
1. Draw and label a picture/diagram. Make a list of all your variables and what they
represent.
2. Which swimmer will take the longest to make the trip? In order to answer this
questions consider what would happen as v’ approaches the speed of the
swimmer, c.
The swimmer that swims upstream will always lose. The best they can hope to
do is tie. If we look at the extremes, when the river is not moving then t = t’. If
we look at the case when c = v’ then the swimmer that swims upstream will
never finish. There is no possible situation in which he is able to swim faster
than the swimmer that swims across the river.
3. Using the equation time = distance/velocity write equations for both t and t’.
2𝑑
𝑡=
𝑐
𝑑
𝑑
′
𝑡 =
+
𝑐 − 𝑣′ 𝑐 + 𝑣′
𝑑(𝑐 + 𝑣 ′ )
𝑑(𝑐 − 𝑣 ′ )
𝑑𝑐 + 𝑑𝑣 ′ + 𝑑𝑐 − 𝑑𝑣 ′
=
+
=
(𝑐 − 𝑣 ′ )(𝑐 + 𝑣 ′ ) (𝑐 − 𝑣 ′ )(𝑐 + 𝑣 ′ )
(𝑐 − 𝑣 ′ )(𝑐 + 𝑣 ′ )
𝒕′ =
𝟐𝒅𝒄
− 𝒗′𝟐
𝒄𝟐
4. Find the ratio t/t’. Show all your work and express your answer in the
simplest/most elegant form possible.
𝑡 2𝑑 𝑐 2 − 𝑣 ′2 𝑐 2 − 𝑣 ′2 𝑐 2 𝑣 ′2
=
∗
=
= 2− 2
𝑡′
𝑐
2𝑑𝑐
𝑐2
𝑐
𝑐
𝒕
𝒗′𝟐
=
𝟏
−
𝒕′
𝒄𝟐
5. In two or more sentences describe what this race looks like to the coach if he is
standing on the shore.
The swimmer swimming across appears to swim in diagonally across the river
and diagonally back.
The swimmer swimming up/down the river appear to swim very slowly
upstream and quickly down stream
6. In two or more sentences describe what this race looks like to the coach if they
are sitting in a canoe floating down the river.
The swimmer swimming across the river swims straight across and straight
back.
The swimmer swimming up/down the river appears to swim a long
ways upstream and then not very far down stream.