Particles are scattered classically by a potential: V (r) = U(1 − r 2/a2 ), for r ≤ a 0, for r > a , U is a constant. Assume that U > 0. A particle of mass m is coming in from the left with initial velocity v0 and impact parameter b < a. Hint: work in coordinates (x, y) not (r, φ). 1. What are the equations of motion for determining the trajectory x(t) and y(t) when r < a? 2. Assume that at t = 0 the particle is at the boundary of the potential r = a. Solve your equations from the previous part to find the trajectory x(t) and y(t) for the time period when r < a. Express your answer in terms of sinh and cosh functions. 3. For initial energy 1 2mv2 0 = U, find the scattering angle θ as function of b.

An charge with mass m and charge q is emitted from the origin, (x,y)=(0,0). a large, flat screen is located at x=l. there is a target on the screen at y position y(h), where y(h) > 0. in this problem, you will examine two different ways that the charge might hit the target. ignore gravity in this problem. 1.assume that the charge is emitted with velocity v(0) in the positive x direction. between the origin and the screen, the charge travels through a constant electric field pointing in the positive y direction. what should the magnitude e of the electric field be if the charge is to hit the target on the screen? express your answer in terms of m, q, y(h), v(0), and l. 2.now assume that the charge is emitted with velocity v(0) in the positive y direction. between the origin and the screen, the charge travels through a constant electric field pointing in the positive x direction. what should the magnitude e of the electric field be if the charge is to hit the target on the screen? express your answer in terms of m, q, y(h), v(0), and l.