Shooting a spaceship with a photon

[E92M3]

Homework Statement

Relative to the rest frame, the trajectory of a spaceship is:
$$x=vt$$
$$y=y_0$$

An observer at:
$$(x,y)=(0,0)$$
wants to send a photon to hit the spaceship at:
$$(x,y)=(0,y_0)$$

a) When and in what direction must the observer send the photon?
b) In the frame of the spaceship, what is the angle between the spaceship and the photon's velocity?

Homework Equations

Basic velocity to distance relations. Possibly Lorentz transformation?

The Attempt at a Solution

\
a) Well, the observer is trying to send a photon from (x,y)=(0,0) to (x,y)=(0,y₀); therefore he muse send it in the +y-direction. The time it takes the photon to reach (x,y)=(0,y₀) is:
$$t=\frac{y_o}{c}$$
Since the spaceship is moving in the +x-direction at speed v, the observer must send the photon when the spaceship is at:
$$x=vt=\frac{-vy_o}{c}$$

b)This part is hard and I have made a few arguments that lead to different paths:

1. The speed of the spaceship in the spaceship's frame is zero, then how could there be an angle?
2. When the photon strikes the spaceship, the photon only has a y velocity while the spacecraft has only a x velocity. Therefore, the angle is 90 degrees.

Which one of my argument is valid?

 

[ideasrule]

I don't think I'm understanding the question. So the spaceship starts at x=0, y=y0, and you want to send a photon there from (0,0)? In order for it to arrive at y=y0 when the spaceship still hasn't moved, the photon will have to travel at infinite speed.

 

[diazona]

I don't think I'm understanding the question. So the spaceship starts at x=0, y=y0, and you want to send a photon there from (0,0)? In order for it to arrive at y=y0 when the spaceship still hasn't moved, the photon will have to travel at infinite speed.

My understanding is that the spaceship can be considered to have been moving at constant speed for all time, and we choose a coordinate system so that the point t=0,x=0,y=y0 is on the spaceship's world line. The photon would have to be emitted at some negative time, as E92M3 found.

E92M3, for part b, first look at argument 2: that holds in "the rest frame," but not in the spaceship's frame. Remember that just because an object has $v_x = 0$ in one frame, does not mean that it has $v_x = 0$ in all frames. So argument 2 is not valid.

In argument 1, you are correct that the spaceship has no velocity in its own rest frame, so you're right that there cannot be an angle between the spaceship's velocity and the photon's velocity in the spaceship's rest frame. However, the problem asks for the angle between the spaceship and the photon's velocity... this is not particularly clear wording, I'll admit, but think about it this way: let's say that the nose of the spaceship is aligned along its velocity in "the rest frame." The spaceship's nose then defines a direction in any frame, whether the spaceship is moving in that frame or not. The problem wants you to find the angle between the spaceship's nose and the photon's velocity, in the spaceship's rest frame.

 

[E92M3]

I don't think I'm understanding the question. So the spaceship starts at x=0, y=y0, and you want to send a photon there from (0,0)? In order for it to arrive at y=y0 when the spaceship still hasn't moved, the photon will have to travel at infinite speed.

That was not specified in the question.

So should I try relativistic velocity addition?