Spacecraft Traveling Past Space Station: Length and Radio Signal Timing

[catalyst55]

A spacecraft (Lo = 80 m) travels past a space
station at speed 0.7c. Its radio receiver is on
the tip of its nose. The space station sends a
radio signal the instant the tail of the spacecraft
passes the space station.

(a) What is the length of the spacecraft in the
reference frame of the space station?
L=80(1-.7^2)^.5 = 57m (fine)

(b) What is the time taken for the radio signal
to reach the nose of the spacecraft ,
according to those on the space station?

t= s/v = 57/c = 1.9x10^-7 s (They get 6.3x10^-7 s (consequential 'errors' result) which can be obtained by t= 57 / 0.3c - but this logic seems to contradict einstein's 2nd postulate since the signal must move relative to the spacecraft at c)



(c) How far from the space station is the nose
of the spacecraft when it receives the radio
signal from the reference frame of the
space station?

S(t) = s(moved) + s(length of ship)

s= 0.7c *(1.9x10^-7) + 57 = 97m
their answer: 190m

(d) What is the time taken for the radio signal
to reach the nose, according to those on
the spacecraft ?

t = s / v = 97 / c = 3.2x10^-7s

their answer: 2.67x10^-7s

What am i doing wrong?

Thanks.

 

[Andrew Mason]

A spacecraft (Lo = 80 m) travels past a space
station at speed 0.7c. Its radio receiver is on
the tip of its nose. The space station sends a
radio signal the instant the tail of the spacecraft
passes the space station.

(a) What is the length of the spacecraft in the
reference frame of the space station?
L=80(1-.7^2)^.5 = 57m (fine)

Correct.

(b) What is the time taken for the radio signal
to reach the nose of the spacecraft ,
according to those on the space station?

t= s/v = 57/c = 1.9x10^-7 s (They get 6.3x10^-7 s (consequential 'errors' result) which can be obtained by t= 57 / 0.3c - but this logic seems to contradict einstein's 2nd postulate since the signal must move relative to the spacecraft at c)

But in the time it takes the light signal to move 57 m. the rocket ship has moved.

You have to use the Lorentz transformation here to determine the time in the space station frame:

$$t = \frac{t' + \frac{vx'}{c^2}}{\sqrt{1 - v^2/c^2}}$$

where $t' = 80/c$ and $x' = 80$

$$t = \frac{80/c + \frac{.7c*80}{c^2}}{\sqrt{1 - .7^2}}$$ = 190/c sec.

(c) How far from the space station is the nose
of the spacecraft when it receives the radio
signal from the reference frame of the
space station?

S(t) = s(moved) + s(length of ship)

s= 0.7c *(1.9x10^-7) + 57 = 97m
their answer: 190m

Again, use the Lorentz transformation:

$$x = \frac{x' + vt'}{\sqrt{1-v^2/c^2}}$$

$$x = \frac{80+.7*80/c}{\sqrt{1-.7^2}}$$

$$x = 190 m.$$

(d) What is the time taken for the radio signal
to reach the nose, according to those on
the spacecraft ?

t = s / v = 97 / c = 3.2x10^-7s

their answer: 2.67x10^-7s

What am i doing wrong?

Again, use the Lorentz transformation to relate space time coordinates in one frame to the other.

$$t' = \frac{t - \frac{vx}{c^2}}{\sqrt{1 - v^2/c^2}}$$

where $t = 190/c$ and $x = 190$$$t' = \frac{190/c - \frac{.7c*190}{c^2}}{\sqrt{1 - .7^2}}$$ = 80/c sec. = 2.67e-7 sec.

AM

 

[Curious3141]

A spacecraft (Lo = 80 m) travels past a space
station at speed 0.7c. Its radio receiver is on
the tip of its nose. The space station sends a
radio signal the instant the tail of the spacecraft
passes the space station.

(a) What is the length of the spacecraft in the
reference frame of the space station?
L=80(1-.7^2)^.5 = 57m (fine)

No problems there.

(b) What is the time taken for the radio signal
to reach the nose of the spacecraft ,
according to those on the space station?

t= s/v = 57/c = 1.9x10^-7 s (They get 6.3x10^-7 s (consequential 'errors' result) which can be obtained by t= 57 / 0.3c - but this logic seems to contradict einstein's 2nd postulate since the signal must move relative to the spacecraft at c)

You're considering everything from the space station's perspective. Let's say the pulse hits the nose of the craft at time t. In that time t, the light pulse would've traveled a distance ct from the space station. The tail of the craft would've traveled 0.7ct from the station. Since the craft is 57 meters long from the perspective of the station,

ct = 0.7ct + 57

t = 57/(0.3c) = 6.3*10^(-7)s, as required.

(c) How far from the space station is the nose
of the spacecraft when it receives the radio
signal from the reference frame of the
space station?

S(t) = s(moved) + s(length of ship)

s= 0.7c *(1.9x10^-7) + 57 = 97m
their answer: 190m

Look at what the answer for b) above. The required distance is simply ct, which is :

d = ct = (c)(57/0.3c) = 57/0.3 = 190 meters.

(d) What is the time taken for the radio signal
to reach the nose, according to those on
the spacecraft ?

t = s / v = 97 / c = 3.2x10^-7s

their answer: 2.67x10^-7s

The simplest way to work this out is to consider that from the perspective of the craft's occupants, the light pulse is traveling at c towards the nose. The distance to be covered is L0 = 80 meters.

Hence, t = 80/c = 2.67*10^(-7)s

A more complicated way to do it would be to transform the time found in b) with the Lorentz transform, but this is needlessly complicated. And if you use the intermediate results, there's a significant round off error.