Question

Physics 4A: Collected Homework 1 1. A proton travels directly (head on) towards a stationary lead nucleus, with an initial speed vo, when the proton is a distance zo from the nucleus. It undergoes an acceler ation of magnitude a = kx-2 directed away from the nucleus, where x is the distance from the proton to the nucleus and k is a positive constant. (Assume the lead nucleus remains stationary throughout.) (a) Give an expression for how the velocity of the proton depends on x. (It will also depend on k, vo, and ro.) (b) Find an expression for the distance of closest approach of the proton to thenu cleus.

Answer 1

Net force n proton q_1 = e q_2 = ze q_2 = 8ze F_net = -Kq_1q_2/x^2 + k/n^2 m a = -e (8ze)/4 pi epsilon_0 x^2 = k/n^2 m dv/dx = 1/x^2 [- 8ze^2/4 pi epsilon_0 + k] m dv = dx/x^2 [k - 8z e^2/4 pi epsilon_0] m integral^v_v_0 dv = [k - 8z e^2/4 pi epsilon_0] integral^n_x_0 dx/x^2 m [v - v_0] = [k - 8z e^2/4 pi epsilon_0] [- 1/x]^k_x_0 m [v - v_0] = [k - 8z e^2/4pi epsilon_0] [1/x_0 - 1/x] v = 1/m [k - 8z e^2/4 pi epsilon_0] [1/x_0 - 1/x] + v_0 use conservation energy to find closest approach - r = closest approach final velocity v = 0 1/2 m v_0^2 + 1/4 pi epsilon_0 ze^2/x_0 = 1/4 pi epsilon_0 ze^2/r + 0 1/r = 4 p I epsilon_0/z e^2 [m v^2_0/2 + 1/4 p I epsilon_0 z e^2/x_0] = 4 pi epsilon_0 m v^2_0 /2 z e^2 + 1/x_0