A rod of proper length L moves past you a speed v. You reach out and grab the back end of the rod, bringing that point instantly to rest in your frame of reference. Assuming that this information, that the back of the rod has stopped, travels toward the front of the rod at the speed of light (it actually travels at the speed of sound), 

(a) Show that for any v > 0 the length of the rod always extends beyond the proper length L_p before the front of the rod comes to rest and the rod assumes its proper length. 

(b) Defining a “coefficient of extension” Δ as 1/L_p times the difference between its maximum length and its proper length, plot a graph of Δ versus v/c for the following values v/c: 0, 0.10, 0.25, 0.40, 0.50, 0.65, 0.80, 0.85, 0.90, 0.95, 0.98. 

(c) What is the maximum length the rod can attain as v → c?