Find the final velocity of each block if the collision is elastic?

Question

A 200g block moves to the right at a speed of 100 cm/sec and meets a 400g block moving to the left with a speed of 80 cm/sec. Find the final velocity of each block if the collision is elastic?

Answer 1

I'll assume right is positive. I'll also convert to SI units to avoid potential unit issues.

An elastic collision means kinetic energy is conserved:

E_K1 = E_K2

(1/2)m₁v_1i² + (1/2)m₂v_2i² = (1/2)m₁v_1f² + (1/2)m₂v_2f²

(1/2)(0.2)(0.1)² + (1/2)(0.4)(-0.08)² = (1/2)(0.2)v_1f² + (1/2)(0.4)v_2f²

0.001 + 0.00128 = 0.1v_1f² + 0.2v_2f²

0.00228 = 0.1v_1f² + 0.2v_2f²

v_1f² + 2v_2f² = 0.0228

Also, momentum is conserved:

P1 = P2

m₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f

(0.2)(0.1) + (0.4)(-0.08) = (0.2)v_1f + (0.4)v_2f

0.02 - 0.032 = 0.2v_1f + 0.4v_2f

-0.012 = 0.2v_1f + 0.4v_2f

-0.12 = 2v_1f + 4v_2f

v_1f + 2v_2f = -0.06 or v_1f = -0.06 - 2v_2f

Now substitute "-0.06 - 2v_2f" in place of "v_1f" in the first equation to get:

v_1f² + 2v_2f² = 0.0228

(-0.06 - 2v_2f)² + 2v_2f² = 0.0228 and solve. Then sub the results into v_1f = -0.06 - 2v_2f to get the other velocity