To determine how long the spacecraft appears to be to an observer on Earth, we need to utilize the concept of length contraction from Einstein's theory of special relativity. Length contraction states that an object moving at a significant fraction of the speed of light will appear shorter to a stationary observer than it does to an observer moving with the object.
The formula for length contraction is given by:
[tex]\[ L = L_0 \sqrt{1 - \frac{v^2}{c^2}} \][/tex]
Where:
- [tex]\( L \)[/tex] is the contracted length (the length observed by someone on Earth).
- [tex]\( L_0 \)[/tex] is the proper length (the length measured by the astronaut inside the spacecraft).
- [tex]\( v \)[/tex] is the velocity of the moving object relative to the observer.
- [tex]\( c \)[/tex] is the speed of light.
Given values:
- Proper length [tex]\( L_0 = 100 \)[/tex] meters.
- Velocity [tex]\( v = 0.85c \)[/tex].
- Speed of light [tex]\( c \)[/tex] (though the exact numerical value of [tex]\( c \)[/tex] is unnecessary due to the velocity being given as a fraction of [tex]\( c \)[/tex]).
Let's now plug in the given values into the length contraction formula.
1. Substituting [tex]\( v = 0.85c \)[/tex] into the formula:
[tex]\[ L = 100 \sqrt{1 - (0.85)^2} \][/tex]
2. Calculate the square of the velocity fraction:
[tex]\[ (0.85)^2 = 0.7225 \][/tex]
3. Subtract this result from 1:
[tex]\[ 1 - 0.7225 = 0.2775 \][/tex]
4. Take the square root of 0.2775:
[tex]\[ \sqrt{0.2775} \approx 0.527 \][/tex]
5. Finally, multiply this result by the proper length [tex]\( L_0 \)[/tex]:
[tex]\[ L = 100 \times 0.527 \approx 52.7 \text{ meters} \][/tex]
Thus, to someone on Earth, the spacecraft appears to be approximately 52.7 meters long. This shorter length is due to the relativistic effects of traveling at a high velocity close to the speed of light.
