3. A particle of mass m initially at rest falls to the center of the field...

Potential (u) = - a/r so, force, (F) = - partial/partial r = a partial/partial r (1/r) or, F = - a/r^2 or, V dv/dr = - a/r^2 or, V dv = - a/r^2 dr or v^2/2 = a/r + c applying initial condition that at r = R, V = 0 so, theta = a/R + c impies C = - a/R therefore r^2/2 = a/r - a/r or V = squareroot 2 a squareroot 1/r - 1/R or dr/dt = squareroot 2 a squareroot 1/r - 1/R or dr/dt = squareroot 2a squareroot R - r/rR or squareroot rR/R - r dr = squareroot 2a dt integrating integral^0_R squareroot rR/R - r dr = squareroot 2 a integral^t_0 dt Let r = R sin^2 theta implies dr = 2 R sin theta cos theta d theta limits: At r = R, theta = pi/2 r = 0, theta = 0 So, 2 R^3/2 integral^0_pi/2