A particle of mass m moves under the influence of a central force.The potential energy function is V(r) = { K (a2âr2) /2 if r ⤠a 0 if r > a
as illustrated in the figure. The initial position is (x,y) = (a/2,0) and the initial velocity is (vx,vy) = (0, v0). [Data: m = 1.0 kg; K =17.1 N/m; a = 1.0 m; r0 = 0.5 a; v0 = 2.0m/s.]
(A) Determine the time ta when the mass crosses the boundary r =a.
(B) Determine angular position Ïa where the particle crosses theboundary r = a.
***HINT GIVEN***
Use the constants of the motion to analyze the orbit.
Write L = mr^2 dÏ/dt = the initial value. What is the initial value?
Write E = (1/2)m (dr/dt)^2 + L^2/(2mr2) +V(r ) = the initial value. What is the initial value?
Part A. In the energy equation, change the variable from r to u, where r =sqrt(u). The result can be solved by u(t) = c1+c2 cosh(αt). Determine c1, c2, and α.