Homework 2 Solutions
Section 2.2 Problem 46) In the theory of relativity, the mass of a
particle whith velocity v is
m0
m= p
1 − v 2 /c2
(1)
where m0 is the mass of the particle at rest and c is the speed of light.
What happens as v → c− .
Solution: We want to know what happens to equation 1 as
v → c− . This is the same as considering the limit limv→c− m(v).
Where we are taking the left handed limit of this particular
mass function. So what happens. As v → c, then v 2 /c2 →, so
the denominator is getting smaller and smaller. But m0 is
constant, so mass become infinite, as v → c.
Q.E.D.
Section 2.3 Problem 54) In the theory of relativity, the Lorentz
contraction formula
p
L = L0 1 − v 2 /c2
(2)
expressed the length L of an object as a function of its velocity v with
respect to an observer, where L0 is the length of the object at rest and c is
the speed of light. Find limv→c− L(v) and interpret the result. Why is the
left-handed limit necessary?
Solution: As you can imagine I chose some problems that are
in common. That is, I chose phisics problems. In doing these
problems you should see something, we do not need to know
phisics to do physics. There is a quote from a great
mathematician that comes to mind: ”Physics is too important to
be left in the hands of physisists.” -David Hilbert. Though I will
not agree or deny his statment it is gratifying that we can do
these kinds of problems whith out study of mathematics.
So the solution. limv→c− L(v) = 0, look at the reasoning in the
above problem. The interesting thing here is why we have to look
1
at the left handed limit. There are two reasons...well only one but
I will take both. First off, the universal maximum speed is c the
speed of light so we cannot approach the speed of light from a
greater speed. The mathematical answer is more or less why c is
the maximum speed in the universe. What happens if v > c, then
1 − v 2 /c2 is nagative, and that it problematic when taking square
roots, so we cannot approach c from the right for this reason.
Also, you should notice something very strange here. When you
travel faster, not only do you appear to be getting heavier, but,
you also appear to be getting smaller. This is actually how the
universe works. You can experience this. Next time you are on
the turnpike and you see a sign to your right speed up and when
you pass it it will appear shorter then it actually is (part of this
is an optical illusion but it gets the point accross). Hope you
enjoyed.
Q.E.D.
2