A car which has a mass of m, needs 27 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 6[math: v] ?*Hint: work out the energies at each velocity separately.

Question

BlackTom AI · Accepted Answer

Kinetic energy is (1/2) m v^2, so the energy depends on the square of the speed.

1) Relevant knowledge points and concepts
- Kinetic energy (KE) formula: KE = (1/2) m v^2.
- If speed is multiplied by a factor k, KE is multiplied by k^2 because of the v^2 dependence.

2) Step-by-step reasoning
- Start with the given: at speed v, KE1 = (1/2) m v^2 = 27 kJ.
- For speed 6v, plug into the KE formula: KE2 = (1/2) m (6v)^2.
- Use algebra to relate KE2 to KE1.

3) Calculations with intermediate steps
- (6v)^2 = 36 v^2.
- KE2 = (1/2) m · 36 v^2 = 36 · [(1/2) m v^2].
- Substitute KE1 = (1/2) m v^2 = 27 kJ:
  KE2 = 36 × 27 kJ.
- Compute 36 × 27:
  36 × 20 = 720,
  36 × 7 = 252,
  720 + 252 = 972.
- So KE2 = 972 kJ.

4) Why the correct answer is correct
- Because kinetic energy scales with the square of speed, increasing speed by a factor of 6 increases energy by 6^2 = 36. Multiplying the initial 27 kJ by 36 gives 972 kJ, matching the correct option.

5) Why the other choices are incorrect (brief)
- a. 486 kJ: equals 27 × 18, not 27 × 36; no physical justification (factor 18 is incorrect).
- b. 162 kJ: equals 27 × 6, which would be correct only if KE scaled linearly with speed (it does not).
- d. 81 kJ: equals 27 × 3, which has no relation to the v^2 scaling required.

6) Conclusion
- The correct energy required at 6v is 972 kJ, so the correct choice is c. 972 kJ.

A car which has a mass of m, needs 27 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 6[math: v] ?*Hint: work out the energies at each velocity separately.单项选择题

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