Suppose Prof. B buys a ladder that is 12.0 m long (as measured when it is at res

Question

Suppose Prof. B buys a ladder that is 12.0 m long (as measured when it is at rest) and wants to use length contraction to make it fit in his 10.0 m long garage (as measured when it is at rest). The garage has a front and a back door so that the ladder can fly straight through.

(a) Assume Prof. L runs with the ladder at speed 0.7c relative to the garage, in the direction parallel to its length. How long will the ladder be according to Prof. B who is standing beside the garage?

(b) How long will the garage be according to Prof. L?

(c) Draw a space-time diagram for Prof. B’s reference frame (S). Use a horizontal scale such that 1 cm on your diagram corresponds to 1 m and a vertical scale such that 3 cm on your diagram corresponds to 10 ns. Draw worldlines for the front and back of the garage (left and right sides), such that the front of the garage is at x = 0 m. With a different colour pen or pencil, draw worldlines for the two ends of the ladder (left and right ends) which are moving in the positive x direction. Let the left end of the ladder pass through the origin. Does the ladder fit in the garage in Prof. B’s frame? What does the word ’fit’ mean in this context?

(d) On your diagram, label the event when the left end of the ladder meets the front (left) side of the garage ’event A’. Label the event when the right end of the ladder meets the back (right) side of the garage ’event B’. What are the space- time coordinates of these two events in Prof. B’s frame? How much time has elapsed between these two events in Prof. B’s frame?

(e) Add space and time axes for Prof. L’s frame (S) to your space-time diagram, (do not attempt to assign a scale to either axis). What are the slopes of the ct and x axes on your ct-x diagram?

(f) Use the Lorentz transformations to find the space-time coordinates for events A and B in Prof. L’s frame (S). Which event occurs first and by how much time in Prof. L’s frame? Use this result to add a scale to the t axis

. (g) Draw a line that passes through event B that is parallel to the x axis. Use the results of part f to add a scale to the t axis. Use the proper length of the ladder to add a scale to the x axis. What do you notice about the relative scale of the axes in each frame on your diagram?

Explanation / Answer

a) using length contraction

we know that L0=12m

v=.7c

then L=Losqrt(1-v^2/c^2)=12*sqrt(1-.7c^2/c^2)=12*.714=8.569m

b) now propre garage length Lo=10m

using length contraction we get

L=10*sqrt(1-.7c^2/c^2)=7.14m

c)

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