Elastic collisions conserve both momentum and kinetic energy

Steps:

• Calculate total momentum before and after: for equal masses, $v_1 + v_2 = u_1 + u_2$.
• Calculate total kinetic energy before and after: $v_1^2 + v_2^2 = u_1^2 + u_2^2$ (masses cancel).
• For option D, before: 4.0 m/s and 1.0 m/s (momentum sum 5.0, KE sum 17); after: 1.0 m/s and 4.0 m/s (momentum sum 5.0, KE sum 17)—both conserved.
• Verify other options fail at least one conservation law.

Why D is correct:

• Velocities exchange, conserving both momentum ($m(4.0 + 1.0) = m(1.0 + 4.0)$) and KE ($\frac{1}{2}m(16 + 1) = \frac{1}{2}m(1 + 16)$), per elastic collision definition for equal masses.

Why the others are wrong:

• A: After KE sum (assuming 2.0 m/s and implied velocity) is 10.25, not 16.25—KE not conserved.
• B: After momentum sum (assuming both 0.5 m/s) is 1.0, not 4.0—momentum not conserved.
• C: After momentum sum is 5.0, not 2.5—momentum not conserved.

Final answer: D