Suppose a space vehicle with a rest mass of 150,000 kg travels past the International Space Station at a constant speed of [tex]2.6 \times 10^8 \, \text{m/s}[/tex] with respect to the I.S.S. An observer on the I.S.S. measures the moving vehicle's length as 25.0 m.

Determine:
1. The relativistic mass of the space vehicle.
2. The length of the space vehicle as measured by an astronaut on the space vehicle.

A. Relativistic mass: 151,364 kg; Length: 25.0 m
B. Relativistic mass: 150,000 kg; Length: 26.2 m
C. Relativistic mass: 152,735 kg; Length: 24.8 m
D. Relativistic mass: 149,500 kg; Length: 25.5 m

The relativistic mass of the space vehicle is approximately 151,364 kg and the length of the space vehicle as measured by an astronaut on board is approximately 25.0 m.

Explanation:

To determine the relativistic mass of the space vehicle, we can use the formula for relativistic mass:

m' = γ * m

where m' is the relativistic mass, γ is the Lorentz factor, and m is the rest mass. The Lorentz factor can be calculated using the formula:

γ = 1 / √(1 - (v^2 / c^2))

where v is the velocity of the space vehicle and c is the speed of light. Plugging in the values, we can find that the relativistic mass is approximately 151,364 kg.

To determine the length of the space vehicle as measured by an astronaut on the space vehicle, we can use the formula for length contraction:

L' = L / γ

where L' is the length as measured by the astronaut and L is the length as measured by the observer on the ISS. Plugging in the values, we can find that the length is approximately 25.0 m.