Two hypothetical planets of masses m1 and m2 and radii r1 and r 2 , respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. (a) When their center-to-center separation is d, find expressions for the speed of each planet and their relative velocity. (b) Find the kinetic energy of each planet just before they collide, if m1 2.00 1024 kg, m2 8.00 1024 kg, r1 3.00 106 m, and r 2 5.00 106 m. (Hint: Both energy and momentum are conserved.)

(a)

(b) Kinetic Energy of planet with mass m₁, is KE₁ =  1.068×10³² J

Kinetic Energy of planet with mass m₂, KE₂ =  2.6696×10³¹ J

Explanation:

Here we have when their distance is d apart

Energy is given by

Conservation of linear momentum gives

m₁·v₁ = m₂·v₂

From which

v₂ =  m₁·v₁/m₂

At equilibrium, we have;

      which gives

multiplying both sides by m₂/m₁, we have

Such that v₁ =

Similarly, with v₁ =  m₂·v₂/m₁, we have

From which we have;

and

The relative velocity = v₁ + v₂ =

v₁ + v₂ =

(b) The kinetic energy KE =

Just before they collide, d = r₁ + r₂ = 3×10⁶+5×10⁶ = 8×10⁶ m

= 10333.696 m/s

=2583.424 m/s

KE₁ = 0.5×2.0×10²⁴× 10333.696² =  1.068×10³² J

KE₂ = 0.5×8.0×10²⁴× 2583.424² =  2.6696×10³¹ J.

he made 11 calls in one day