Verified answer

Energy conservation law for comet: E1+W1 = E2+W2, where E is kinetic energy at the distance x, W is potential energy at the distance x; let W1 =0, then

0.5*m*v1^2 = 0.5*m*v2^2 +W2, where m is comet mass;

elementary potential energy dW = {GM*m/x^2}*dx, where GM*m/x^2 is gravitational force between comet and Sun, G is gravitational constant, M is the mass of Sun;

thence W2 = GM*m* integral{for x=x1 until x2} of dx/x^2 = -1/x {for x=x1 until x2} = GM*m* (1/x1 –1/x2);

thence v2 = sqrt(v1^2 -2GM * (1/x1 –1/x2)); on the other hand the centripetal force at x1 F(x1)= GM*m/x1^2 = m*(v1^2)/x1 or GM = (v1^2)*x1

and v2 = v1* sqrt(2x1/x2 –1); (Kepler if I’m not mistaken)

notice: cross products [x1*v1] = [x2*v2] would be also correct, but not just x2v2.

Use conservation of angular momemtum of the second body in the central force of the sun.

a) x1v1 = x2v2 -> v2 = v1*x1/x2

b) Plug in x2 as 5.28X10^9 km