Question

· I. A rocketship of length 100 m when at rest, traveling at u/c = 0.6, carries a radio receiver at its nose. A radio pulse is emitted from a stationary space station just as the tail of the rocket passes the station. (a) From the point of view of the rocket reference frame, how far from the tail does the radio pulse move before being detected by the receiver in the nose? What is the velocity of the radio pulse in the reference frame of the rocket? (b) From the point of view of the rocket reference frame, what is the time interval between the the emission of the radio signal at the rocket's tail to it's detection at the rocket's nose? (c) From the point of view of the space station reference frame, how far from the space station is the nose of the rocket at the instant of arrival of the radio signal at the nose? Hint: Lorentz transformations, using previous results, can be very helpful (d) By space-station time, what is the time interval between the arrival of this signal and its emission from the station? . 2. (Based on a question from the text by Tipler) In Episode 5 of Star Wars, the Empire's spaceships launch probe droids throughout the galaxy to seek the base of the Rebel Alliance. Suppose a spaceship moving at 2.3 × 108 m/s toward Hoth (site of the rebel base) launches a probe droid toward Hoth at 2.1 × 108 ml/s relative to the spaceship - (a) According to Galilean Relativity what is the speed of the probe droid relative to Hoth? (b) In terms of special relativity if rebel astronomers are watching the approaching spaceship through as telescope, will they see the probe before it lands on Hoth? (c) According to Special Relativity, what is the speed of the probe droid relative to hoth?

just I need

ans for 1 (a)

and 2(c)

don't need remaining please help me in solving the problems

Answer 1

(1) In case of special relativity the length contraction in the direction of motion is given by

Where, L is contraction in length

L₀ is proper length of the object

v is speed of the object

C = 3 x 10⁸ m/s is speed of light

According to the question,

L₀ = 100 m

v = 0.6 C

(a)

Let the signal starts at x = x' = 0 and time t = t' = 0  

And the signal arrives at the nose of rocket. let the position is x = x₂ and time is t = t₂ .

In space station frame (i.e. S frame) the rocket is contracted. So, the nose of rocket is at

So, in the space station frame the initial position of nose was at x = L = 80 m and the nose is moving with 0.6 C.

And the signal moves with C. And its initial position was at x = 0 .

Hence,

or,   

or,   

Hence the signal will travel 200 m before being detected.

And the velocity of radio pulse will be C (According to the second postulate of Special theory of relativity).

(2) In the case of relativistic speed the velocity addition formula is

  

Where, $u_{1}$ is speed of first object (direction will be given by + or - sign)

$u_{2}$ is speed of second object

   is speed of light

Given,

speed of spaceship = 2.3 x 10⁸ m/s

speed of probe droid = 2.1 x 10⁸ m/s

Both speed is towards the hoth .

So, speed of probe relative to hoth will be

or,   

or,   

For any doubt please comment and please give an up vote. Thank you.