A car which has a mass of m, needs 32 kJ of energy to travel at a velocity of [math: v]. How much energy is needed for the same car to travel at a speed of 5[math: v] ?*Hint: work out the energies at each velocity separately.

Question

BlackTom AI · Accepted Answer

Kinetic energy depends on the square of speed: KE = 1/2 m v^2.

1. Relevant knowledge points and concepts
- Kinetic energy (KE) of a mass m moving at speed v is KE = 1/2 m v^2.
- If speed is multiplied by a factor k, the kinetic energy is multiplied by k^2 because of the v^2 dependence.
- Given KE at v is 32 kJ, we can use proportional reasoning to find KE at 5v.

2. Step-by-step reasoning
- Let KE_v denote the energy at speed v: KE_v = 1/2 m v^2 = 32 kJ (given).
- For speed 5v, KE_5v = 1/2 m (5v)^2.
- Expand the square: (5v)^2 = 25 v^2, so KE_5v = 1/2 m * 25 v^2.

3. Calculations with intermediate steps
- KE_5v = 25 * (1/2 m v^2).
- Substitute KE_v = 32 kJ: KE_5v = 25 * 32 kJ.
- Multiply: 25 * 32 = (25 * 30) + (25 * 2) = 750 + 50 = 800.
- Therefore KE_5v = 800 kJ.

4. Why the correct answer is correct
- Because kinetic energy scales with the square of speed, increasing the speed by a factor of 5 increases energy by 5^2 = 25. Multiplying the given 32 kJ by 25 yields 800 kJ, matching the physics formula exactly.

5. Why the other options are incorrect
- a. 160 kJ: this is 5 times 32 kJ (linear scaling), incorrect because KE scales with the square, not linearly.
- b. 80 kJ: this is 2.5 times 32 kJ, no physical basis here.
- c. 400 kJ: this is 12.5 times 32 kJ (half of the correct factor), also incorrect.

6. Conclusion
- The required energy for speed 5v is 800 kJ, so the correct choice is d. 800 kJ, which matches the provided answer.

A car which has a mass of m, needs 32 kJ of energy to travel at a velocity of [math: v]. How much energy is needed for the same car to travel at a speed of 5[math: v] ?*Hint: work out the energies at each velocity separately.Single choice

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