Answer
See explanation.
Work Step by Step
The astronaut lies along the direction of the rocket’s motion, so his height is Lorentz-contracted.
a. The doctor in the rocket measures the proper length of 2.0 m.
The value of $\gamma$ is $\frac{1}{\sqrt{1-u^{2}/c^{2}}}=\frac{1}{\sqrt{1-0.910^2}}=2.412$.
The person on earth measures a height of $\frac{2.00m}{\gamma}= 0.829 m$.
b. If the contracted height had been 2.00 m, the doctor on board would have measured $(2.00m)\gamma=4.82m$. This is not reasonable.
c. 2.00m for both. There is no length contraction in a direction perpendicular to the motion.
