A stationary particle explodes into 3: A (to the left), B and C (both to the right). B has mass m and speed 3v. C has mass 2m and speed v. A has speed 2v. What is the mass of A in terms of m?

The key to solving this is remembering that momentum is conserved. The large, initial particle has no speed so its momentum is zero. Therefore, if we add together the momenta of the final particles we also get zero. So we can write:

p_A + p_B + p_C = 0

And we can rearrange for p_A, which is what we want to find:

p_A = -p_B - p_C

We know that momentum is calculated p = mv and we are given the masses and velocities of B and C, and the velocity of A (we remember that A is travelling in the opposite direction so has a negative v):

M_A * (-2v) = -3mv - 2mv

We rearrange for the mass of A, M_A, and find that:

M_A = 2.5 m