A moving object has kinetic energy. When work is done on it from outside, its energy changes.
Basic Knowledge
Kinetic energy K:K = ½mv²
m is mass(kg), and v is speed(m/s). The unit is J.
When speed doubles, energy becomes four times as large(it is proportional to v²).
Work-energy theorem:Total work done by external forces = change in kinetic energyW_net = ΔK = ½mv² - ½mv₀²
Positive work done on an object increases its kinetic energy, while negative work decreases it.
This theorem can be derived directly from Newton's equation of motion.
Kinetic energy K (kinetic energy)（½mv². Energy possessed by a moving object）
Work-energy theorem（Work done by the net force = change in kinetic energy）
W_net = ΔK（Total work = final K - initial K）
Energy (energy)（The ability to do work. Unit: J）
½mv²（One-half the product of mass and the square of speed）
Double speed → four times K（Because it is proportional to v²）
Further Study (Background and Significance)
The work-energy theorem provides a powerful method: if force and distance are known, speed can be calculated. The three equations of uniformly accelerated motion(in particular v² = v₀² + 2ax) are actually transformations of the work-energy theorem:
From F = ma, W = Fx = max. ΔK = ½mv² - ½mv₀² = m(v²-v₀²)/2 = max. These agree.
The work-energy theorem is the starting point for the idea that “when a force does work, energy is transferred.” When friction is present, the negative work done by friction represents conversion into thermal energy.
The damage caused by highway accidents is proportional to the square of speed because kinetic energy is proportional to v². Doubling the speed makes the energy received in a collision(= damage) four times as large.
- K = ½mv²(proportional to the square of speed)
- Double speed → four times K
- Work-energy theorem: W_net = ΔK
- Positive work → increase in K(acceleration)
- Negative work → decrease in K(deceleration)
- When multiple forces act, use the work done by the net force
- Unit: J = kg·(m/s)²
Points to Note (Common Confusions)
① Do not forget the ½ in K = ½mv². K = mv² is incorrect. ② Use speed(a scalar), which is the magnitude of velocity(a vector). ③ The work-energy theorem uses the work done by the net force(the sum of all forces). The work done by only one force is insufficient. ④ “Energy increased” means “positive work was done,” and “energy decreased” means “negative work was done.”
Practice
- An object of mass 2 kg is moving at 6 m/s. Find its kinetic energy.
- A work of 80 J was done on an object of mass 4 kg that was initially at rest. Find its final speed.
- An object of mass 3 kg moving at 10 m/s had 24 J of work done on it by friction(negative work). Find its final speed.