Galilean Transformations Problem: Two moving Rockets+Missile

[Shadowmaker]

Homework Statement

Mary and Frank are each in their own rocket ships moving along the x-axis. Mary's ship passes Frank's ship at t₀=t₀'=0 with a speed "v" to the right. When t=t₁ in Frank's frame, Frank shoots a missile with a speed "u" where u>v in the direction of Mary. At time t=t₂ in Frank's frame the missile hits Mary's ship.

a. Show the process of events on an x-t diagram showing Frank, Mary, and the missiles position as a function of time.
Given [t₁, u, v ] Determine,
b. The slope and y intercept of Mary's motion on the diagram
c. The slope and y intercept of the missile on the diagram
d. From the two lines, calculate the time t₂ and the positions x₁ and x₂ of the missile in Frank's frame
e. Using Galilean transformations, determine the time t₁' and t₂' of the events in Mary's frame.
f. Using Galilean transformations, determine the positions x₁' and x₂' of the missile in Mary's frame.

*Everything here is written exactly as it is on the sheet. Didn't change a thing.

Homework Equations

t=t'
x'=x-vt
y'=y
x=x₀+v₀t+1/2at²
v=v₀+at
u_x'=u_x-v
u_y'=u_y

The Attempt at a Solution

This problem has me very confused and I am not sure if the wording is off or if I'm simply not understanding.
Mary passes Frank with a speed of v relative to what?
In Frank's frame he fires a missile at speed u which is apparently greater than v but to Frank, Mary wouldn't be traveling at v right? She'd be traveling at v minus Frank's speed.

Ignore what I believe to be an inconsistency, I drew this picture of their positions with respect to time from an observer's frame.

and this one from Frank's frame

to b) and c) I wrote that Mary's slope is v and the missiles slope is u, while both their y-intercepts are 0 because they're both moving along the x-axis. (I'm unsure if by y-intercepts it actually means t-intercepts)

but I get stuck on d), e), and f) using either method

edit-----------------------
for d) I did the following
From Frank's frame
x₁ = 0 for the missile.

u(t₂-t₁) = vt₂
isolating for t₂ gets t₂ = ((ut₁)/(u-v))

and

x₂ = ((u²t₁)/(u-v)) - t₁

for e) I used the formula above simply saying t₁'=t₁ and t₂ = t₂

and f) got me
x'=x-vt
x₁' = x₁ -vt₁. x₁ = 0
x₁' = -vt₁.

x₂'=x₂-vt₂
x₂' = ((u²t₁)/(u-v)) - t₁ - v((ut₁)/(u-v))

I feel like its totally wrong

 

[Gene Naden]

Hi Shadowmaker,
The wording seems OK. You asked, "speed v relative to what?" This is the speed of Mary relative to Frank.
Yes, Mary is traveling to the right.
In Frank's frame, Frank's speed is zero of course.
Your first diagram appears correct. However, I believe it could be simpler; You could use Frank's frame as the observer, in which Frank's speed is zero and so his line would vertical. I see you did this on your second diagram and this looks right to me.

The y intercept of course means when t=0, as I think you understand. Mary's y intercept in Frank's frame would be at the origin, which you have correctly shown in your second diagram.

The "y intercept" of the missile, in Frank's frame, would be at t=0 which is before the missile is fired. So I agree with you that it seems more reasonable to talk about the t intercept, which is when x=0.

Regarding part d, your equation t2 = (ut1)/(u-v) seems right. I notice that if u=v, t2 is infinity. This means that the missile never hits Mary's ship, which is of course correct.

However, your equation x2 = ((u2t1)/(u-v)) - t1 cannot be correct because x2 is a distance, while the second term on the right, t1, is a time.

Thanks,
Gene

 

[Shadowmaker]

Hi Shadowmaker,
The wording seems OK. You asked, "speed v relative to what?" This is the speed of Mary relative to Frank.
Yes, Mary is traveling to the right.
In Frank's frame, Frank's speed is zero of course.
Your first diagram appears correct. However, I believe it could be simpler; You could use Frank's frame as the observer, in which Frank's speed is zero and so his line would vertical. I see you did this on your second diagram and this looks right to me.

The y intercept of course means when t=0, as I think you understand. Mary's y intercept in Frank's frame would be at the origin, which you have correctly shown in your second diagram.

The "y intercept" of the missile, in Frank's frame, would be at t=0 which is before the missile is fired. So I agree with you that it seems more reasonable to talk about the t intercept, which is when x=0.

Regarding part d, your equation t2 = (ut1)/(u-v) seems right. I notice that if u=v, t2 is infinity. This means that the missile never hits Mary's ship, which is of course correct.

However, your equation x2 = ((u2t1)/(u-v)) - t1 cannot be correct because x2 is a distance, while the second term on the right, t1, is a time.

Thanks,
Gene

Thank you very much! I already turned it in, but I believe I had the right answers before I did so, what you’ve written increased my confidence.

I did end up looking for the t intercept ultimately.

I used line the y=mx+b line form with the understand that the slope was actually 1/v and 1/u.

Since Mary’s t intercept was 0, hers was just y=mx.

In frank’s frame X1 = 0.

My last edit was using the mechanics formulas. I wasn’t sure about the time I got but I knew that the x2 was wrong because from mary’s frame, the missle at t2 should be at her 0.

Using the line’s formulas instead I got x1 = 0, t2 = ut1/(u-v)
And for x2 I believe* (as it’s not with me anymore) I got uvt1/(u-v) and when I converted it to x2’ it came out to 0 which is what it should have done.

Thanks!