PHY 3221 Classical Mechanics, Homework #10, due 8:30 am, 04/16/12
1 Consider the event, E1 , located at (x, ct) = (1000, 1000) in a stationary frame, in which lengths
are measured in meters.
a) Calculate the coordinates of this event in a frame moving at 3/5 the speed of light with respect
to the stationary frame.
b) A second frame is moving at −4/5 the speed of light relative to the first moving frame. Using
your answer in part (a), calculate the coordinates of the same event in the second moving frame.
c) A third frame is moving at −5/13 the speed of light relative to the stationary frame. Calculate
the coordinates of event, E1 , in this third frame.
d) Use the addition formula for relativistic velocities to calculate the velocity of the second frame
relative to the stationary frame.
e) Comment on the relation between your answers in parts (b) and (c).
1
2 Consider the following four-vectors,
X1 = (x1 , y1 , z1 , ct1 ) = (200, 300, 600, 700),
X2 = (x2 , y2 , z2 , ct2 ) = (300, 200, 700, 600),
X3 = (x3 , y3 , z3 , ct3 ) = (200, 400, 400, 600),
X4 = (x4 , y4 , z4 , ct4 ) = (200, 400, 400, 700),
where all lengths are measured in meters.
a) Calculate the length of each of these four-vectors, and show whether they are time-like, space-like
or light-like vectors.
b) Calculate the vector dot products X1 · X2 and X1 · X3 and X3 · X4 . Given your results in part
(a), comment upon the fact that none of these dot products is zero.
c) Suppose that X1 + X3 is proportional to the proper four-velocity of a particle. Find the proportionality constant which gives it the correct normalization, and give the γ factor.
d) Why can X2 − X3 not be proportional to a proper four-velocity?
e) Suppose that X4 − X3 is proportional to the proper four-velocity of a particle. Find the proportionality constant which gives it the correct normalization, and describe what sort of four-velocity
it represents (hint: calculate the γ factor again).
2
3 A particle of mass 10 kg moves at 96% the speed of light.
a) Calculate its proper four-velocity.
b) Calculate its proper four-momentum.
c) It interacts with another identical particle moving in the opposite direction at the same speed
and they stick together. What is the mass of the combined object?
d) Suppose, instead, that the particle collides with, and sticks to, an identical particle traveling at
the same speed, but moving at right angles to the first particle (as seen from a stationary frame).
What is the final speed of these joined particles?
e) What is the mass of these joined particles?
3