Two balls with the same kinetic energy have mass of ball a = m and ball b = 2m. What is the ratio of their momentums: a/b?

We know that both balls have the same kinetic energy, so we can write E_ka = E_kb We also know that kinetic energy is given by the equation E_k = 1/2 (mv²) and momentum, p, is p = mv where m is the mass and v is the velocity. Since m is constant, we need an expression for the velocity of ball a, v_a, in terms of the velocity of ball b, v_b. so the kinetic energy for ball a and b respectively is E_ka = 1/2 * m * v_a²    E_kb = 1/2 * 2m * v_b² = m * v_b² as we know, these are equal and so 1/2 * m * v_a² = m * v_b² simplifying and rearranging to find an expression for v_a in terms of v_b we get v_a² = 2 * v_b² ,  v_a = 2^1/2 v_b substituting this into the equation for momentum and finding the ratio of a to b, we get p_a / p_b = m * 2^1/2v_b / 2m * v_b      = 2^1/2 / 2      = 1 / 2^1/2