Answers to Chapter Five Questions
5.1) and 5.2) are shown below on the spacetime
diagram. Note that the time T = 10 ns
falls between T = 6 ns and T = 12 ns so I estimated its location.
5.3) LEarth/Lbus
= 1 inch/100 feet = 1 inch/1200 inches = 1/1200 = the shrinkage factor,
,
where c is taken to be 1. Square both
sides of the equation,
,
and solve for v. 
5.4) t
= 
x + 
where v/c = 3/5 so t = 
x + 
.
When T = 8, t = 
x + 
and when T = 16, t
= x + 
5.5) When
x = 6, the equation for the T = 8 ns line is 
x + = 6 + = 
+ = 
= 10 ns!
Perfect.
When
x = 12, the equation for the T = 16 ns line is x + = 12 + 
= 
+ = 
= 20 ns!
Perfect again.
5.6a) A
constant X line has slope m = 
so it can be represented on an x vs t graph
by,
t
= 
x + 
.
5.6b) tB = 
xB since point B lies on the X-axis
where t = 
.
Replace t and x in the answer for 5.6a) in terms of xB to get, 
xB
= 
+
5.6c) Solve
the equation in 5.6b) for bB, bB = xB
= 
5.6d) XB
= 
xB
5.6e) bB = 
5.6f) t
= x 
,
this is the equation for the line with constant X value, XB.
5.6g) X = 
(x
– vt). This is the equation that gives
the bus coordinate X for any spacetime point with Earth coordinates x and t.
5.7) We are given x = 200 ft and t = 200 ns and
want to find the X and T that correspond to that spacetime point. From the book with v = 3/5 and c = 1, the
equations for X and T become,
and 
x).
Now insert x = 200 ft and t = 200 ns to get,
= 
ft and 
200) = 100 ns just like required.
5.8) For
this problem, the bus coordinates are known, X = 160 ft and T = 0 ns, and the
Earth coordinates are wanted. The
equations with v = 3/5 are given below
and 
X).
= 200 ft and 
160) = 120 ns.
These are exactly the coordinates of Bev when Ed passed.
5.9) u
= 
= 
=
c
5.10) u
= 
= 
=
-c
5.11) 
.
When x = 1/100 = 0.01, 
and
1 - x = 0.990000, the difference is 0.000099!
When x = 1/10000 = 0.0001, 
and
1 - x = 0.9999900000, the difference is 0.0000000001!
When x = 1/1000000 = 0.000001, 
and
1 - x = 0.9999999, and the difference is so small that it did not show up on my
calculator!!
The conclusion is that
the approximation is very good when x is small compared to 1.