Suppose that each of the three masses in the figure below has a mass of 5.05 kg, a radius of 0.0679 m, and are separated by a distance of 9.9 m. If the balls are released from rest, what speed will they have when they collide at the center of the triangle? Ignore gravitational effects from any other objects.

The objects in this question are arranged at the vertexes of an equilateral triangle.

So far the only ways I have found to solve this problem involve calculus but we are expressly forbidden from using and calculus.

I have tried using conservation of energy (.5mv^2 - G(m)(m)/r) but always end up with the final radius as zero, which seems wrong because then I am dividing by zero.

Any help wold be greatly appreciated!

The final radius (distance between particle centers) is not zero. It is twice the radius.

Subtract the final potential energy of the complete system from the initial P.E.

The difference will be the final kinetic energy, which will be evenly divided between the three particles (because of symmetry).

If a is the distance between particle centers in an equilateral triangle configuration, the PE of the system is

-Gm^2/a -2Gm^2/a = -3Gm^2/a

P.E change = 3Gm^2[1/.136 - 1/9.9]

1/3 of that is the KE at collision, per particle.

To find the speed at which the masses collide at the center of the triangle, we can use the principle of conservation of linear momentum. Since the objects are at rest initially, the total momentum before the collision is zero. After the collision, the masses will move in a straight line towards the center, so their momenta will be in the same direction.

To understand how to solve this problem, we can break it down into smaller steps:

Step 1: Calculate the momentum of each mass before the collision.
Since the masses are at rest initially, their initial momentum is zero.

Step 2: Determine the direction and magnitude of the total momentum after the collision.
Since the balls move towards the center of the triangle in a straight line, their momenta will add up to form the total momentum. But since the masses are moving towards each other, they will have opposite directions. So mathematically, the total momentum after the collision is the sum of the magnitudes of individual momenta multiplied by their respective directions.

Step 3: Calculate the velocity of each mass after the collision.
Divide the total momentum after the collision by the mass of each object to get the velocity of each object.

Step 4: Determine the speed at which the masses collide at the center of the triangle.
Since the balls move towards the center, their velocities add up in magnitude. So, to find the speed at which they collide, add the magnitudes of their velocities at the center.

Now, let's proceed with the calculations:

Step 1: Since the masses are initially at rest, their momentum is zero before the collision.

Step 2: The magnitudes of the momenta after the collision are the same for each object because they have the same mass. Therefore, the total momentum after the collision is 2 * (mass of one object) * (velocity of one object).

Step 3: Divide the total momentum after the collision by the mass of each object to get the velocity of each object. Let's call this velocity 'v'.

Step 4: To find the speed at which the masses collide at the center, we add the magnitudes of their velocities at the center. Since the velocities have the same magnitude but opposite directions, we have 2 * v.

By following this approach, we have used only principles of linear momentum and did not require calculus to solve the problem.