Question

Two stars orbit each other, far from other objects. Star 1 has a mass M1 = 5.0 x 10+30 kg. Star 2 has a mass M2 = 3.0 X 10+30 kg. At a certain instant of time, the positions and velocities of the stars are as follows: Star 1 : ~ r1,i = < 2.0, 0.0, 0 > x 10+9 m, ~ v1,i = < 0.0, 4.0, 0 > x 10+6 m/s. Star 2 : ~ r2,i = < -3.3, 0.0, 0 > x 10+9 m, ~ v2,i = < 0.0, -6.7, 0 > x 10+6 m/s. The gravitational constant G = 6.67 X 10-11 Nm2/Kg2.
Q- What is the new center-to-center distance between the stars when their respective velocities are: ~ v1,f = < -4.0, 0.0, 0 > x 10+6 m/s and ~ v2,f = < 6.7, 0.0, 0 > x 10+6 m/s

Answer 1

Energy will remain conserved

Kinetic Energy + Potential Energy = Constant

Now, |r1,i - r2,i| = ri = ((2-(-3.3))² + (0-0)² +(0-0)²)^0.5 = 5.5*10⁹m

Initial potential energy P1 = GM1M2 / ri = 6.67*10^-11 * 5*10³⁰*3*10³⁰ / 5.5 *10⁹ = 18.191 * 10⁴⁰ J

Initial Kinetic Energy K1 = (1/2)M1v1i² +(1/2)M2v2i² = 0.5*5*10³⁰*(4*10⁶)² + 0.5*3*10³⁰*(6.7*10⁶)² = 107.355*10⁴² J

In 2nd case, we see that magnitude of velocities for the masses remain the same. So kinetic energy remains the same.

So potential energy also con't change [From conservation of energy, Kinetic Energy + Potential Energy = Constant]

As gravitationial potential P = GMm/R, => Distance between the stars remains the same.

So distance between centres= ri = 5.5 *10⁹ m