To find the speed of each asteroid just before they collide, we can use the conservation of energy and the conservation of linear momentum.
Let's start with the conservation of energy. At the beginning, both asteroids are at rest, so their initial kinetic energy is zero. As they move towards each other and eventually collide, their potential energy decreases and is converted into kinetic energy.
The total initial potential energy of the system is given by the gravitational potential energy equation:
U_initial = - (G * M * M) / (10R)
Here, G is the gravitational constant.
At the point of collision, all the potential energy is converted into kinetic energy. The total final kinetic energy can be calculated as:
K_final = K1_final + K2_final
where K1_final and K2_final are the kinetic energies of asteroid 1 and asteroid 2, respectively.
Using the conservation of linear momentum, we know that the total initial linear momentum of the system is zero, as both asteroids are initially at rest. At the point of collision, the total final momentum is still zero, as the two asteroids collide and stick together.
The linear momentum of an object is given by:
p = mv
where p is the linear momentum, m is the mass, and v is the velocity.
Let's calculate the initial and final kinetic energies:
K_initial = 0
K_final = 1/2 * m1 * v1_final^2 + 1/2 * m2 * v2_final^2
Setting the initial potential energy equal to the final kinetic energy and solving for v1_final and v2_final, we get:
- (G * M * M) / (10R) = 1/2 * M * v1_final^2 + 1/2 * 2M * v2_final^2
Simplifying the equation, we have:
- (G * M * M) / (10R) = 1/2 * M * v1_final^2 + M * v2_final^2
Now, let's solve for v1_final and v2_final.
Using the conservation of linear momentum, we know that:
m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final
Since both asteroids are initially at rest, we have:
m1 * v1_final + m2 * v2_final = 0
Simplifying this equation, we get:
M * v1_final + 2M * v2_final = 0
Now, we have a system of two equations with two unknowns (v1_final and v2_final):
- (G * M * M) / (10R) = 1/2 * M * v1_final^2 + M * v2_final^2
M * v1_final + 2M * v2_final = 0
Solving these equations simultaneously will give us the values of v1_final and v2_final, which are the speeds of asteroid 1 and asteroid 2 just before they collide.
Please note that expressing the final answers in terms of the variables M, R, and appropriate constants is crucial to obtaining the correct solutions.