28.4 Relativistic Addition of Velocities

Suppose a space probe moves away from the Earth at a speed $0 . \text{350} c$. It sends a radio wave message back to the Earth at a frequency of 1.50 GHz. At what frequency is the message received on the Earth?

How is the relativistic Doppler effect different from the classical Doppler effect?

Suppose a spaceship heading straight towards the Earth at $0 . \text{750} c$ can shoot a canister at $0 . \text{500} c$ relative to the ship. (a) What is the velocity of the canister relative to the Earth, if it is shot directly at the Earth? (b) If it is shot directly away from the Earth?

If a spaceship is approaching the Earth at $0.100 c$ and a message capsule is sent toward it at $0.100 c$ relative to the Earth, what is the speed of the capsule relative to the ship?

If a galaxy moving away from the Earth has a speed of $1000 km/s$ and emits $\text{656 nm}$ light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on the Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of the Earth in its orbit negligible here?

If two spaceships are heading directly towards each other at $0 . \text{800} c$, at what speed must a canister be shot from the first ship to approach the other at $0 . \text{999} c$ as seen by the second ship?

When a missile is shot from one spaceship towards another, it leaves the first at $0 . \text{950} c$ and approaches the other at $0 . \text{750} c$. What is the relative velocity of the two ships?

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

Prove that for any relative velocity $v$ between two observers, a beam of light sent from one to the other will approach at speed $c$ (provided that $v$ is less than $c$, of course).

(a) All but the closest galaxies are receding from our own Milky Way Galaxy. If a galaxy $\text{12} . 0 \times \text{10}^{9} \text{ ly}$ ly away is receding from us at 0.$0.900 c$, at what velocity relative to us must we send an exploratory probe to approach the other galaxy at $0.990 c$, as measured from that galaxy? (b) How long will it take the probe to reach the other galaxy as measured from the Earth? You may assume that the velocity of the other galaxy…