Modern Physics - Problem Set

Question

7. What should be the time and place of events 1 and 2 as observed from a rocket frame moving at a velocity 0.92c relative to the lab frame?

 

8. In a space bus a bullet shoots forward with a speed of 0.75c as measured by an observer in the space bus. The said space bus is moving with a velocity of 0.75c as measured by an observer on Earth. What is the velocity of the bullet as measured by the observer on Earth

 

9. A rocket moving at the velocity of 0.87c along the x-axis in the laboratory frame. In the rocket frame, the coordinates are:

 

x' = 10m

y' = 7m

z' = 3m

ct' = 20m

 

What are the coordinates in the lab frame (x, y, z, t)?

Answer 1

Phoebe,

All the questions can be answered from the Lorentz Transformations (LT) between the 4-coordinates of an event A (x_,recorded by an observer in K and those of the SAME EVENT A as observed by an observer in K' . Let K,K' be two intertidal RF, their origins at t=0 coinciding and their X-directions parallel to each other

If RF K' is moving relative to RF K along the X with Velocity V then, using the notations

β=V/C ;γ=(1-β^2), τ=Cxt the LT has the most symmetric and simple form:

τ(A)=γ^-1[τ(A)'+βx(A)'] (1)

x(A)=γ^-1[x(A)'+βτ(A)':] (2)

y(A)=y(A)' (3)

z(A)=z(A)' (4)

If a moving object A is observed in K and K'

then you can get the LT of velocity X-component (which is the only component you are asked) - derived below (result - in (8))

From (1) and (2)

dτ(A)=γ^-1[dτ(A)'+βdx(A)'] (5)

dx(A)=γ^-1[dx(A)'+βdτ(A)':] (6)

Let's call

β(A)=v(A)/C

β(A)'=v(A)'/C

So, β(A)=V(A)/C=(dx(A)/dτ(A)=[dx(A)'+βdτ(A)']/

[dτ(A)'+βdx(A)'] (7)

We could then formally divide both the numerator and the denominator in (7) dτ(A) to get

β(A)=[β(A)'+β]/[1+ββ(A)'] (8)

What you need to do is to correctly substitute the numbers and get back to the "normal" notations for the coordinates and velocity

t=τ/C ; v(A)=β(A)C; v(A)'=β(A)'C

Pay attention to the directions of v(A)' by using it proper sign (e g. is an object in K' thrown in negative direction of x, use negative sign of v(A)'

Good luck

Dr.Ariel