Deep space resupply. A rocket (frame S′) bound for Planet X leaves Earth (frame S) traveling at speed β = 3/5 in the positive x direction. We will label the departure as Event0,forwhichx=x′ =ct=ct′ =0. Afteroneyearintherocket(t′1 =1yr),the crew realizes that they have used up 10% of their food supplies and that they will run 1 out in exactly nine more years (i.e., ten years from their departure.) They immediately send a distress signal to Earth asking for help (Event 1). The signal travels at the speed of light, and mission control on Earth receives the signal at time t2 (Event 2). In your answers, express time in units of years (yr) and distance in units of light-years (c·yr). (a) Draw two spacetime diagrams (ct(′) vs. x(′)), one in S and one in S′, that show Events 0, 1, and 2. Label the axes, events, worldlines and their slopes. (b) Determine the spacetime coordinates of Events 1 and 2 in both S and S′. Use the Lorentz transformation where useful. (c) Immediately upon receiving the distress signal, Earth sends a message back at the speed of light saying, “More food is on the way!” and also immediately launches an emergency rocket with supplies. At what time t′3 does the crew receive the reassuring message (Event 3)? (d) The emergency rocket reaches the crew at the exact moment their food runs out (Event 4). What was the speed of the rocket in the Earth frame? (e) At what time t4 does the emergency rocket reach the crew at Event 4 according to an observer on Earth? (f) Add Events 3 and 4 to your S-frame spacetime diagram from part (a). Draw the emergency rocket’s worldline and label its slope. (g) Can there be another frame (perhaps the reference frame of an alien observer watching all this from their own fast-moving spaceship) in which the
Deep space resupply. A rocket (frame S′) bound for Planet X leaves Earth (frame S) traveling at speed β = 3/5 in the positive x direction. We will label the departure as Event0,forwhichx=x′ =ct=ct′ =0. Afteroneyearintherocket(t′1 =1yr),the crew realizes that they have used up 10% of their food supplies and that they will run 1 out in exactly nine more years (i.e., ten years from their departure.) They immediately send a distress signal to Earth asking for help (Event 1). The signal travels at the speed of light, and mission control on Earth receives the signal at time t2 (Event 2). In your answers, express time in units of years (yr) and distance in units of light-years (c·yr). (a) Draw two spacetime diagrams (ct(′) vs. x(′)), one in S and one in S′, that show Events 0, 1, and 2. Label the axes, events, worldlines and their slopes. (b) Determine the spacetime coordinates of Events 1 and 2 in both S and S′. Use the Lorentz transformation where useful. (c) Immediately upon receiving the distress signal, Earth sends a message back at the speed of light saying, “More food is on the way!” and also immediately launches an emergency rocket with supplies. At what time t′3 does the crew receive the reassuring message (Event 3)? (d) The emergency rocket reaches the crew at the exact moment their food runs out (Event 4). What was the speed of the rocket in the Earth frame? (e) At what time t4 does the emergency rocket reach the crew at Event 4 according to an observer on Earth? (f) Add Events 3 and 4 to your S-frame spacetime diagram from part (a). Draw the emergency rocket’s worldline and label its slope. (g) Can there be another frame (perhaps the reference frame of an alien observer watching all this from their own fast-moving spaceship) in which the
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Deep space resupply. A rocket (frame S′) bound for Planet X leaves Earth (frame S) traveling at speed β = 3/5 in the positive x direction. We will label the departure as Event0,forwhichx=x′ =ct=ct′ =0. Afteroneyearintherocket(t′1 =1yr),the crew realizes that they have used up 10% of their food supplies and that they will run
1
out in exactly nine more years (i.e., ten years from their departure.) They immediately send a distress signal to Earth asking for help (Event 1). The signal travels at the speed of light, and mission control on Earth receives the signal at time t2 (Event 2).
In your answers, express time in units of years (yr) and distance in units of light-years (c·yr).
(a) Draw two spacetime diagrams (ct(′) vs. x(′)), one in S and one in S′, that show Events 0, 1, and 2. Label the axes, events, worldlines and their slopes.
(b) Determine the spacetime coordinates of Events 1 and 2 in both S and S′. Use the Lorentz transformation where useful.
(c) Immediately upon receiving the distress signal, Earth sends a message back at the speed of light saying, “More food is on the way!” and also immediately launches an emergency rocket with supplies. At what time t′3 does the crew receive the reassuring message (Event 3)?
(d) The emergency rocket reaches the crew at the exact moment their food runs out (Event 4). What was the speed of the rocket in the Earth frame?
(e) At what time t4 does the emergency rocket reach the crew at Event 4 according to an observer on Earth?
(f) Add Events 3 and 4 to your S-frame spacetime diagram from part (a). Draw the emergency rocket’s worldline and label its slope.
(g) Can there be another frame (perhaps the reference frame of an alien observer watching all this from their own fast-moving spaceship) in which the crew runs out of food before the resupply rocket arrives? Explain.
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