Two planets of mass 7 x 10^25 kg are release 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 this distance apart, we need to apply the law of conservation of mechanical energy. At the initial distance, the gravitational potential energy is converted into kinetic energy when the planets come closer.
Let's break down the steps to solve this problem:
Step 1: Calculate the gravitational potential energy at the initial distance.
The gravitational potential energy can be calculated using the formula: U = -(G * m1 * m2) / r,
where U is the gravitational potential energy, G is the gravitational constant (6.67 x 10^-11 N(m/kg)^2), m1 and m2 are the masses of the planets, and r is the initial distance between them.
U = - (6.67 x 10^-11 N(m/kg)^2) * (7 x 10^25 kg)^2 / (2.5 x 10^12 m)
Step 2: Calculate the kinetic energy at the initial distance.
Since the planets are at rest initially, their kinetic energy is zero.
K_initial = 0
Step 3: Calculate the total mechanical energy at the initial distance.
The total mechanical energy is the sum of the gravitational potential energy and the kinetic energy.
E_initial = U + K_initial
Step 4: Calculate the half distance between the planets.
The half distance is half of the initial distance.
half_distance = (2.5 x 10^12 m) / 2
Step 5: Calculate the gravitational potential energy at the half distance.
Using the same formula as before, but substituting the half distance for r.
U_half = - (6.67 x 10^-11 N(m/kg)^2) * (7 x 10^25 kg)^2 / (half_distance)
Step 6: Calculate the kinetic energy at the half distance.
When the planets are half the initial distance apart, all the gravitational potential energy should be converted into kinetic energy.
K_half = U_half
Step 7: Calculate the speed at the half distance.
The kinetic energy can be calculated using the formula: K = (1/2) * m * v^2,
where K is the kinetic energy, m is the mass of the planets, and v is the speed.
K_half = (1/2) * (7 x 10^25 kg) * v^2
Now, equating K_half to U_half, we can solve for v.
(1/2) * (7 x 10^25 kg) * v^2 = U_half
Step 8: Solve for the speed.
To find v, divide both sides of the equation by (1/2) * (7 x 10^25 kg) and take the square root of both sides.
v = √(U_half / [(1/2) * (7 x 10^25 kg)])
Once you substitute the values for U_half and evaluate the expression, you will find the speed of the planets when they are half this distance apart.