An interstellar space probe is launched from Earth. After a briefperiod of accel

Question

An interstellar space probe is launched from Earth. After a briefperiod of acceleration, it moves with a constant velocity,70.0% of thespeed of light. Its nuclear-powered batteries supply the energy tokeep its data transmitter active continuously. The batteries have alifetime of 15.1 years as measured in arest frame. (a) How long do the batteries on the spaceprobe last as measured by mission control on Earth?
1 yr

(b) How far is the probe from Earth when its batteries fail asmeasured by mission control?
2 ly

(c) How far is the probe from Earth as measured by its built-intrip odometer; when its batteries fail?
3 ly

(d) For what total time after launch is data received from theprobe by mission control? Note that radio waves travel at the speedof light and fill the space between the probe and Earth at the timethe battery fails.
4 yr (a) How long do the batteries on the spaceprobe last as measured by mission control on Earth?
1 yr

(b) How far is the probe from Earth when its batteries fail asmeasured by mission control?
2 ly

(c) How far is the probe from Earth as measured by its built-intrip odometer; when its batteries fail?
3 ly

(d) For what total time after launch is data received from theprobe by mission control? Note that radio waves travel at the speedof light and fill the space between the probe and Earth at the timethe battery fails.
4 yr

Explanation / Answer

(a)
   the relation ship between time intervals in the movingand at rest system is
   t = tp
        = tp /[1 - (v / c)2]
        = 15.1 yr / [1 -(0.70)2]
        = ......... yr
(b)
   we know that
   velocity = distance / time
   d = v (t)
= (0.70 c) (t)  
= ........ ly
(c)
   the astronauts see earth flying out the back window at0.70 c
   so we get
   d ' = v (tp)
= (0.70 c) (15.1 yr)  
= ........ ly
(d)
   the mission controls gets signals for (t) whilethe battery is operating and then for d yr after the
   battery stops powering the transmitter d ly away so weget
   d + d' = .......yr

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