Sample Exercise: A Contraction in Length
A spaceship traveling at the velocity of $1.8 × 10^{8}$ m/s (0.6c) covers a distance of 900 km as measured by an observer on the Earth.
a. What is the distance traveled in this time as measured by the pilot of the spaceship?

v = 0.6c
$c = 3 × 10^{8}$ m/s
$L_{0} = 900$ km
L = ?
b. How much time does it take to cover this distance as measured by the observer on Earth and as measured by the pilot?

t = ?
$t_{0} = ?$

Question

Sample Exercise: A Contraction in Length
A spaceship traveling at the velocity of [latex]1.8 × 10^{8}[/latex] m/s (0.6c) covers a distance of 900 km as measured by an observer on the Earth.
a. What is the distance traveled in this time as measured by the pilot of the spaceship?

v = 0.6c
[latex]c = 3 × 10^{8}[/latex] m/s
[latex]L_{0} = 900[/latex] km
L = ?
b. How much time does it take to cover this distance as measured by the observer on Earth and as measured by the pilot?

t = ?

[latex]t_{0} = ?[/latex]

Accepted Answer

From table 20.1,

table 20.1
Values of γ for Different Values of Relative Speed v
v = 0.01c	γ = 1.000 05
v = 0.1c	γ = 1.005
v = 0.5c	γ = 1.155
v = 0.6c	γ = 1.250
v = 0.8c	γ = 1.667
v = 0.9c	γ = 2.294
v = 0.99c	γ = 7.088

γ = 1.25

[latex]\frac{1}{γ }= \frac{1}{1.25 }[/latex]

[latex]\frac{1}{γ }= 0.8[/latex]

[latex]L = (\frac{1}{γ }) L_{0}[/latex]

= (0.8)(900 km)
= 720 km

b.

As seen by the observer on Earth: [latex]L_{0} = vt[/latex]

[latex]t = \frac{L_{0}}{v} [/latex]

[latex]= \frac{9 × 10^{5} m}{1.8 × 10^{8} m/s}[/latex]

[latex]= 5 × 10^{-3} s = 5 ms[/latex]

As seen by the spaceship pilot: [latex]L = v t_{0}[/latex]

[latex]t_{0} = \frac{L}{v}[/latex]

[latex]= \frac{7.2 × 10^{5} m/s}{1.8 × 10^{5} m/s}[/latex]

[latex]= 4 × 10^{-3} s = 4 ms[/latex]

Question 20.2: Sample Exercise: A Contraction in Length A spaceship traveli...

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