The figure shows two train cars that move toward each other, collide, and couple together.

How much thermal energy is created in this collision?

Figure caption: Before and after diagrams show forces involved in a collision between two train cars.

Question

The figure shows two train cars that move toward each other, collide, and couple together.

How much thermal energy is created in this collision?

Figure caption: Before and after diagrams show forces involved in a collision between two train cars.

BlackTom AI · Accepted Answer

Energy converted to thermal energy equals the loss of mechanical kinetic energy: ΔEth = Kinitial − Kfinal (because momentum is conserved but the collision is inelastic).

Relevant knowledge points:
- Conservation of momentum determines the final velocity of the coupled cars (here the problem gives the final speed).
- Kinetic energy is 1/2 m v^2 and is scalar (use squared speeds).
- Thermal energy produced = total initial kinetic energy − total final kinetic energy.

Step-by-step calculation:
1) Write given quantities:
- m1 = 2.0×10^4 kg, v1i = +1.5 m/s
- m2 = 4.0×10^4 kg, v2i = −1.1 m/s
- Combined mass m = m1 + m2 = 6.0×10^4 kg
- Given final speed vf = −0.25 m/s (direction irrelevant for kinetic energy)

2) Compute initial kinetic energies:
- K1 = 1/2 m1 v1i^2 = 0.5 × (2.0×10^4 kg) × (1.5 m/s)^2
  = 1.0×10^4 × 2.25 = 22,500 J
- K2 = 1/2 m2 v2i^2 = 0.5 × (4.0×10^4 kg) × (1.1 m/s)^2
  = 2.0×10^4 × 1.21 = 24,200 J
- Total initial K = K1 + K2 = 22,500 J + 24,200 J = 46,700 J

3) Compute final kinetic energy of the coupled cars:
- Kfinal = 1/2 (m1 + m2) vf^2 = 0.5 × (6.0×10^4 kg) × (0.25 m/s)^2
  = 3.0×10^4 × 0.0625 = 1,875 J

4) Thermal energy generated:
- ΔEth = Kinitial − Kfinal = 46,700 J − 1,875 J = 44,825 J

Why the correct answer is correct:
This collision is inelastic (cars couple), so kinetic energy is not conserved; the lost kinetic energy appears as thermal (and deformation) energy. Subtracting final from initial kinetic energy gives 44,825 J released as thermal energy.

Why other options are incorrect:
- 1,875 J is the final kinetic energy, not the lost (thermal) energy.
- 46,700 J is the total initial kinetic energy, not the energy converted.
- 24,200 J and 22,500 J are the individual initial kinetic energies of one car each, not the thermal energy.

Final confirmation: the thermal energy created is 44,825 J, matching the provided correct answer.

The figure shows two train cars that move toward each other, collide, and couple together. How much thermal energy is created in this collision? Figure caption: Before and after diagrams show forces involved in a collision between two train cars.Single choice

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