If the two protons and the are all at rest after the collision, find the initial speed of the protons.

To find the initial speed of the protons, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.

The momentum of an object is given by the product of its mass and velocity. Since both protons are at rest before the collision, their initial momentum is zero.

After the collision, if the two protons and the unnamed particle are all at rest, their total momentum is still zero. This means that the total momentum before the collision is equal to the total momentum after the collision, which is zero.

Now, we have to consider the momentum of the unnamed particle after the collision. Let's assume its mass is denoted as m and its final velocity is v. The momentum of the unnamed particle after the collision is mv.

Since the total momentum before the collision is zero, we can write the equation:

0 = mv

Since m ≠ 0, we can divide both sides of the equation by m:

0/m = v
0 = v

This means that the velocity of the unnamed particle after the collision is 0 m/s.

Since the two protons are initially at rest and the unnamed particle is at rest after the collision, their total initial momentum is also zero. This means that the total initial momentum of the protons is zero.

The momentum of a single proton is given by the product of its mass (m) and its initial velocity (u). Since their total initial momentum is zero, we can write the equation:

0 = mu + mu

Simplifying the equation:

0 = 2mu

Since 2m ≠ 0, we can divide both sides of the equation by 2m:

0/2m = u
0 = u

Therefore, the initial speed of the protons is 0 m/s.