Work, power and energy

Why does a car with the accelerator pressed to the floor still have a top speed? How can you find a skier's speed at the bottom of a slope without knowing its shape? Both answers come from one idea: forces transfer energy by doing work. The CSCA syllabus line «Work and energy, law of conservation of mechanical energy» is exactly this, and energy ideas return later in electricity, heat and modern physics. You will need Newton's second law from the lesson «Newton's laws of motion and their applications»; the next lesson, «Momentum, impulse and conservation of momentum», combines energy with momentum.

Work of a constant force

The sign of the work is decided by cos α. For 0 ≤ α < 90° the work is positive: the force helps the motion and gives the body energy. For α = 90° the work is zero. For 90° < α ≤ 180° the work is negative: the force resists the motion and takes energy away; we then say that the body does work against this force (Chinese 克服…做功).

Power and a car at constant power

A car engine works at most at its rated power P. Since P = Fv, a higher speed means a smaller traction force F. With a resistive force f the acceleration a = (F − f)/m keeps falling while the car speeds up, and the speed stops growing when F has dropped to f. That is the car's maximum speed.

In the second standard case (恒定加速度启动) the car first accelerates uniformly with a constant traction force F = f + ma. The power P = Fv reaches the rated power at v₁ = P/F: this is the end of the uniform acceleration, not the maximum speed. After that the car goes on at constant power with a decreasing acceleration up to v_max = P/f.

Kinetic energy and the work–energy theorem

The theorem follows from Newton's second law and v² − v₀² = 2as: multiply F = ma by s and you get Fs = mv²/2 − mv₀²/2. Its big advantage: neither the time nor the acceleration is needed, and the path can be split into stages with different forces — just add the works of all stages.

1. 1
   Choose the body and two states

   Fix the start and the end; a body at rest has Ek = 0.

2. 2
   List the forces

   Write the work of each force with its sign: positive, negative or zero. Treat each stage with its own forces.

3. 3
   Write the theorem

   W₁ + W₂ + … = Ek₂ − Ek₁ and solve for the unknown.

4. 4
   Check

   Units and sign: distances and forces must come out positive; the mass often cancels.

Potential energy and conservation of mechanical energy

The sum E = Ek + Ep is the mechanical energy (机械能). Gravity and spring forces only move energy between the kinetic and potential forms, so if no other force does work, this sum stays constant.

Friction and heat: when mechanical energy is not conserved

When friction, air resistance or a pushing force does work, the mechanical energy changes. Chinese textbooks state this as the work–energy relation (功能关系): the work of all forces other than gravity and spring forces equals the change of mechanical energy. The mechanical energy lost to kinetic friction becomes heat.

Key terms and how CSCA asks about this

In the Chinese version of the test the same ideas appear under the names below. Even if you take the test in English, recognising them will help you at a Chinese university.

Questions on this topic usually follow one of these patterns; you have about 75 seconds for each:

• the work of a force at an angle, or the total work of several forces (watch the signs!);
• average vs instantaneous power; the power of gravity on a falling body; lifting at constant speed (P = mgv);
• a car at rated power: v_max = P/f, the acceleration at a given speed, the end of uniform acceleration;
• the work–energy theorem: stopping distance, average resistive force, «a push, then sliding»;
• conservation of mechanical energy: a smooth track, a pendulum, a spring launcher, the height where Ek = Ep; graphs of Ek and Ep against height (two straight lines crossing at H/2);
• heat from friction: lost mechanical energy, a conveyor belt, Q = f · s_rel.