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A firecracker explodes in reference frame S at $\mathrm{t}=1.0\mathrm{s}$. A second firecracker explodes at the same position at $\mathrm{t}=3.0\mathrm{s}$. In reference frame S鈥�, which moves in the x-direction at speed v, the first explosion is detected at $\mathrm{x}\text{'}=4.0\mathrm{m}$and the second at $\mathrm{x}\text{'}=-4.0\mathrm{m}$.

a. What is the speed of frame S鈥� relative to frame S?

b. What is the position of the two explosions in frame S?

FIGURE Q36.1 shows two balls. What are the speed and direction of each (a) in a reference frame that moves with ball 1 and (b) in a reference frame that moves with ball 2?

You are standing at $x=9.0km$and your assistant is standing a $x=3.0km$Lightning bolt $1$strikes at$x=0km$and a lightning bolt $2$strikes at$x=12.0km$. You see the flash from bolt $2$at$t=10ms$and the flash from bolt$1att=50ms$ According to your assistant, were the lightning strikes simultaneous? If not, which occurred first, and what was the time difference between the two?

You are standing at $x=9.0km.$Lightning bolt $1$strikes at $x=0km$and lightning bolt $2$strikes at $x=12.0km.$ Both flashes reach your eye at the same time. Your assistant is standing at $x=3.0km.$ Does your assistant see the flashes at the same time? If not, which does she see first, and what is the time difference between the two?

You are flying your personal rocket craft at $0.90c$from Star $A$toward Star $B.$ The distance between the stars, in the stars鈥� reference frame, is $1.0ly.$ Both stars happen to explode simultaneously in your reference frame at the instant you are exactly halfway between them. Do you see the flashes simultaneously? If not, which do you see first, and what is the time difference between the two?

A cosmic ray travels $60km$ through the earth鈥檚 atmosphere in $400渭s$ as measured by experimenters on the ground. How long does the journey take according to the cosmic ray?

At what speed, as a fraction of $c,$does a moving clock tick at half the rate of an identical clock at rest?

An astronaut travels to a star system $4.5ly$away at a speed of $0.90c.$Assume that the time needed to accelerate and decelerate is negligible.

a. How long does the journey take according to Mission Control on earth?

b. How long does the journey take according to the astronaut?

c. How much time elapses between the launch and the arrival of the first radio message from the astronaut saying that she has arrived?

a. At what speed, as a fraction of $c,$must a rocket travel on a journey to and from a distant star so that the astronauts age $10$years while the Mission Control workers on earth age $120$years?

b. As measured by Mission Control, how far away is the distant star?

At what speed, as a fraction of $c,$ would a round-trip astronaut 鈥渓ose鈥� $\frac{1}{25}$ of the elapsed time shown on her watch?