Question

Space Probe #1 passes very close to Earth at a time that both we (on Earth) and the onboard computer on Probe 1 decide to call t = 0 in our respective frames. The probe moves at a constant speed of 0.5c away from Earth. When the clock aboard Probe 1 reads t = 60 sec, it sends a light signal straight back to Earth.

a) At what time was the signal sent, according to the earth’s rest frame?

b) At what time in the earth’s rest frame do we receive the signal?

c) At what time in Probe 1’s rest frame does the signal reach earth?

d) Space Probe #2 passes very close to earth at t = 1 sec (earth time), chasing Probe 1. Probe 2 is only moving at 0.3c (as viewed by us). Probe 2 launches a proton beam (which moves at v = 0.21c relative to Probe 2) directed at Probe 1. Does this proton beam strike Probe 1? Please answer twice, once ignoring relativity theory, and then again using Einstein!

Answer 1

a)

Given that clock on probe1 reads 60 sec,

The speed of the probe is 0.5c away from earth.

Moving clocks and clocks at rest are related by ( time dilation),

where,

: Lorentz factor

v: speed of probe 1

c: speed of light

is the time at which the signal was sent according to earth's frame.

b)

The time taken by the light signal to reach earth is the sum of time at which signal was sent from probe1 and the time taken for the light signal to travel the distance from the position when the signal left probe 1 to earth.

is the time in earth's frame when the light signal reaches earth.

c)

It's a similar calculation as above, but done in the probe1 reference system

is the time in probe1 frame when the light signal reaches earth.

d)

For the proton beam to strike probe1, the distance travelled by both the proton beam and the probe 1 should be equal at a particular moment(moment when the beam strikes).

Ignoring special relativity the velocities add up in the normal way.

Velocity of proton beam = 0.3c + 0.21c = 0.51c

Distance travelled by beam =

Velocity of probe1 = 0.5c

Looking at the velocities of probe1 and beam, our insight tells us that there is a chance of beam striking the probe 1 because

Distance travelled by probe 1 =

Equating

This tells us that, the beam strikes probe1 at 50 seconds.

Now considering special relativity,

Velocity of probe1 = 0.5c

Distance travelled by probe 1 =

Velocities add up using lorentz transformations,

Dividing these both,

Is the speed of the beam.

Now speed of beam< speed of probe 1, we might guess that the beam cannot strike the probe1.

Equating

Time taken to travel cannot be negative, this means the proton beam cannot strike probe1 as we've guessed earlier.