Question

Two spacecrafts A and B approach the moon from opposite directions with speeds of
2.2 × 108 ms−1 and 2.5 × 108 ms−1, respectively, as measured by an observer on the
moon. Calculate the speed of A with which it approaches the moon as observed by
an observer in B.

Answer 1

Answer on Question #60712, Physics / Other

Two spacecrafts A and B approach the moon from opposite directions with speeds of $2.2 \times 10^{8}$ ms$^{-1}$ and $2.5 \times 10^{8}$ ms$^{-1}$, respectively, as measured by an observer on the moon. Calculate the speed of A with which it approaches the moon as observed by an observer in B.

Solution:

The Lorentz velocity transformation:

$$u_{x}' = \frac{u_{x} - v}{1 - u_{x} v / c^{2}}$$

where $u_{x}$ is the velocity of an object measured in the S frame, $u_{x}'$ is the velocity of the object measured in the S' frame and $v$ is the velocity of the S' frame along the $x$ axis of S.

We take the S frame to be attached to the moon and the S' frame to be attached to spacecraft B moving with velocity $v = -2.5 \times 10^{8} \, \text{ms}^{-1}$ along the x axis. Spacecraft A has velocity $u_{x} = 2.2 \times 10^{8} \, \text{ms}^{-1}$ in S.

It follows from first equation that spacecraft A has velocity

$$u_{x}' = \frac{2.2 \cdot 10^{8} + 2.5 \cdot 10^{8}}{1 + 2.2 \cdot 10^{8} \cdot 2.5 \cdot 10^{8} / (3 \cdot 10^{8})^{2}} = 2.92 \cdot 10^{8} \, \text{m/s}$$

Answer:

Spacecraft A moves with velocity $2.92 \cdot 10^{8} \, \text{m/s}$ as measured by an observer in spacecraft B.

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