Two planets of mass 7 x 10^25kg are released from rest 2.5 x 10^12 m apart. What is the speed of the planets when they are half this distance apart?

To find the speed of the planets when they are half the initial distance apart, we can use the principle of conservation of mechanical energy. Assuming there are no external forces acting on the planets, their total mechanical energy will remain constant throughout their motion.

The total mechanical energy of the system is the sum of the gravitational potential energy and the kinetic energy. At the initial distance, the planets are at rest, so their initial kinetic energy is zero. Therefore, the total mechanical energy is equal to their initial gravitational potential energy.

The gravitational potential energy between two objects is given by the formula:

GPE = (G * m1 * m2) / r

where G is the gravitational constant (6.67 x 10^-11 N * (m/kg)^2), m1 and m2 are the masses of the objects (in this case, both planets have the same mass, so we will call it "m"), and r is the distance between the objects.

Using this formula, we can calculate the initial gravitational potential energy when the planets are 2.5 x 10^12 m apart:

GPE_initial = (G * m^2) / r_initial

Next, we can find the final distance between the planets when they are half the initial distance apart. Given that the initial distance is 2.5 x 10^12 m, the final distance will be:

r_final = (1/2) * r_initial

Substituting this value into the gravitational potential energy formula, we can calculate the final gravitational potential energy.

GPE_final = (G * m^2) / r_final

Since the total mechanical energy is conserved, we have:

GPE_initial = GPE_final

Setting the two expressions for gravitational potential energy equal to each other, we can solve for the mass (m):

(G * m^2) / r_initial = (G * m^2) / r_final

Simplifying the equation, we find that the mass cancels out:

1 / r_initial = 1 / r_final

Substituting the values for the initial and the final distances, we can solve for the final distance:

1 / (2.5 x 10^12) = 1 / (1/2 * 2.5 x 10^12)

Simplifying, we get:

1 / (2.5 x 10^12) = 2 / (2.5 x 10^12)

Now we have the final distance between the planets.

Finally, to find the speed of the planets at the final distance, we can use the conservation of mechanical energy again. The total mechanical energy at the final distance will be the sum of the final gravitational potential energy and the final kinetic energy. Since the planets are now moving, their kinetic energy will be non-zero.

The formula for kinetic energy is:

KE = (1/2) * m * v^2

where m is the mass of the objects (in this case, the mass of one planet), and v is the speed.

To find the speed, we can set the total mechanical energy equal to the sum of the gravitational potential energy and the kinetic energy:

GPE_final + KE_final = GPE_initial

Substituting the values for gravitational potential energy and solving for the speed (v), we can calculate the speed of the planets at half the initial distance.