Inelastic collision: apply momentum conservation to find final velocity, then compute KE difference.

Steps:

• Assign directions: carriage 1 velocity +2.00 m/s, carriage 2 velocity -1.00 m/s (towards each other).
• Initial momentum: (5000 kg)(2.00 m/s) + (5000 kg)(-1.00 m/s) = 10,000 kg m/s - 5,000 kg m/s = 5,000 kg m/s.
• Final velocity (joined mass 10,000 kg): v = total momentum / total mass = 5,000 kg m/s / 10,000 kg = 0.50 m/s.
• Initial KE: (1/2)(5000)(2.00)^2 + (1/2)(5000)(1.00)^2 = 10,000 J + 2,500 J = 12,500 J. Final KE: (1/2)(10,000)(0.50)^2 = 1,250 J. KE lost: 12,500 J - 1,250 J = 11,250 J.

Why C is correct:

• Matches exact calculation of initial minus final KE using conservation of momentum in perfectly inelastic collision.

Why the others are wrong:

• A: Equals final KE, ignores initial KE.
• B: Underestimates loss by using only one initial KE term.
• D: Equals total initial KE, ignores final KE.

Final answer: C