To determine which spacecraft appears the longest to an outside observer, we need to consider the relationship between an object's speed and its observed length.
For an outside observer, the length of a moving spacecraft appears shortened due to the effect of Lorentz contraction, a relativistic phenomenon described by the formula:
[tex]\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \][/tex]
where:
- [tex]\( L \)[/tex] is the observed (contracted) length
- [tex]\( L_0 \)[/tex] is the proper (rest) length
- [tex]\( v \)[/tex] is the speed of the spacecraft
- [tex]\( c \)[/tex] is the speed of light
The greater the speed [tex]\( v \)[/tex], the more significant the length contraction. Conversely, the slower the spacecraft moves, the less it contracts, and thus the longer it appears.
Given the percentages of the speed of light ([tex]\( c \)[/tex]) at which the spacecraft are traveling:
- Spacecraft W: [tex]\( 0.95c \)[/tex]
- Spacecraft X: [tex]\( 0.40c \)[/tex]
- Spacecraft Y: [tex]\( 0.10c \)[/tex]
- Spacecraft Z: [tex]\( 0.69c \)[/tex]
First, we convert these percentages into actual speeds:
- Speed [tex]\( c \)[/tex] is [tex]\( 6 \times 10^7 \, \text{m/s} \)[/tex]
- Spacecraft W: [tex]\( 0.95c = 0.95 \times 6 \times 10^7 = 5.7 \times 10^7 \, \text{m/s} \)[/tex]
- Spacecraft X: [tex]\( 0.40c = 0.40 \times 6 \times 10^7 = 2.4 \times 10^7 \, \text{m/s} \)[/tex]
- Spacecraft Y: [tex]\( 0.10c = 0.10 \times 6 \times 10^7 = 6 \times 10^6 \, \text{m/s} \)[/tex]
- Spacecraft Z: [tex]\( 0.69c = 0.69 \times 6 \times 10^7 = 4.14 \times 10^7 \, \text{m/s} \)[/tex]
Next, we identify the spacecraft with the smallest velocity to find which one appears the longest due to the least length contraction:
- Spacecraft W: [tex]\( 5.7 \times 10^7 \, \text{m/s} \)[/tex]
- Spacecraft X: [tex]\( 2.4 \times 10^7 \, \text{m/s} \)[/tex]
- Spacecraft Y: [tex]\( 6 \times 10^6 \, \text{m/s} \)[/tex]
- Spacecraft Z: [tex]\( 4.14 \times 10^7 \, \text{m/s} \)[/tex]
The smallest actual speed is that of Spacecraft Y: [tex]\( 6 \times 10^6 \, \text{m/s} \)[/tex].
Therefore, the spacecraft that appears the longest to an outside observer is Spacecraft [tex]\( \textbf{Y} \)[/tex], because it travels at the slowest speed and experiences the least amount of relativistic length contraction.
