3. A spaceship (S1) is approaching Earth at a speed 3c/5, measured by an observer on...

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3. A spaceship (S1) is approaching Earth at a speed 3c/5, measured by an observer on Earth. A second spaceship (S2) is also approaching Earth at a speed 4c/5, but from the opposite direction as depicted in the figure below. At t = 0, S1 is at a distance ₁ from Earth, whereas S2 is at 12. Both spaceships are travelling parallel to the z-direction. At t = 0, a light pulse of wavelength 500 nm measured on S1 is sent (still parallel to the x-direction) towards Earth from S1. Spaceship S1 X₂ EARTH X₂ Spaceship S2 (a) Calculate the speed of the spaceship S2 according to an observer on S1. (b) Using the fact that we can connect 4-momenta measured in different frames via a Lorentz transformation, derive an equation for the wavelength of the light pulse measured by an observer on Earth and calculate its numerical value. (c) Using the fact that we can connect 4-momenta measured in different frames via a Lorentz transformation, derive an equation for the angle a beam of light sent from Earth along the y-direction (according to the observer on Earth) is seen by the observer on S2. (d) The two spaceships arrive on Earth simultaneously at t = 50 s. On a space- time diagram show the worldlines of the spaceships, Earth and of the photon sent from S1, as seen by the observer on Earth. Note: numerical values are expected to be placed along the r and ct coordinates. (e) The observer on Earth performs an experiment in which a 7-photon of wavelength 0.01 nm is scattered by a stationary electron (Compton effect). The wavelength of the scattered photon increases by 10%. Calculate the energies of the 7-photon and electron before and after collision. →→X 3. A spaceship (S1) is approaching Earth at a speed 3c/5, measured by an observer on Earth. A second spaceship (S2) is also approaching Earth at a speed 4c/5, but from the opposite direction as depicted in the figure below. At t = 0, S1 is at a distance ₁ from Earth, whereas S2 is at 12. Both spaceships are travelling parallel to the z-direction. At t = 0, a light pulse of wavelength 500 nm measured on S1 is sent (still parallel to the x-direction) towards Earth from S1. Spaceship S1 X₂ EARTH X₂ Spaceship S2 (a) Calculate the speed of the spaceship S2 according to an observer on S1. (b) Using the fact that we can connect 4-momenta measured in different frames via a Lorentz transformation, derive an equation for the wavelength of the light pulse measured by an observer on Earth and calculate its numerical value. (c) Using the fact that we can connect 4-momenta measured in different frames via a Lorentz transformation, derive an equation for the angle a beam of light sent from Earth along the y-direction (according to the observer on Earth) is seen by the observer on S2. (d) The two spaceships arrive on Earth simultaneously at t = 50 s. On a space- time diagram show the worldlines of the spaceships, Earth and of the photon sent from S1, as seen by the observer on Earth. Note: numerical values are expected to be placed along the r and ct coordinates. (e) The observer on Earth performs an experiment in which a 7-photon of wavelength 0.01 nm is scattered by a stationary electron (Compton effect). The wavelength of the scattered photon increases by 10%. Calculate the energies of the 7-photon and electron before and after collision. →→X