Work, Energy and Power — Formulas & Examples

Work, energy, and power describe different parts of the same story: work is energy transferred by a force through a displacement, energy is how much can change in a system, and power is how fast that transfer happens. The one check that clears up most beginner mistakes is that work is a transfer, energy is an amount, and power is a rate.

The Core Formulas And Their Symbols

For a constant force acting over a displacement,

where $\theta$ is the angle between the force and the displacement. If the force points along the motion, the work is positive; against the motion, negative; perpendicular to the motion, zero. In mechanics, two common energy formulas are

The first is kinetic energy. The second is the change in gravitational potential energy near Earth's surface. The reference level for potential energy is chosen by you, but the change $\Delta U_g$ is what matters in most problems. Average power and instantaneous power are

The instantaneous form is useful only when you mean force and velocity at the same moment.

Why These Quantities Are Linked

Work is the bridge between force and energy, and the link is not arbitrary. Start from Newton's second law: a net force changes an object's velocity. When you accumulate that net force over the distance the object moves, you are adding up $F\,d$ contributions, and that running total is exactly the change in the quantity $\tfrac{1}{2}mv^2$. That is why net work equals the change in kinetic energy:

If the net work is positive, kinetic energy increases; if negative, it decreases. Power then drops out naturally: it is just how fast that work accumulates in time. Two machines can do the same amount of work, but the one that finishes in less time has greater power. So the three ideas are one chain: force builds work, work changes energy, and power is the rate of that change.

Worked Example: Lifting A Backpack

Suppose you lift a $10\,\mathrm{kg}$ backpack straight upward by $2\,\mathrm{m}$ in $4\,\mathrm{s}$ at roughly constant speed. Because the speed is approximately constant, the upward force you apply is approximately equal to the backpack's weight:

The force and displacement point in the same direction, so $\theta = 0$ and $\cos\theta = 1$. The work you do on the backpack is

That same lift increases the backpack's gravitational potential energy by

So the work you do becomes gravitational potential energy. The average power is

The backpack gains $196\,\mathrm{J}$, and you transfer that energy at an average rate of $49\,\mathrm{J/s} = 49\,\mathrm{W}$. One subtle point: the net work on the backpack is approximately zero because your positive work is balanced by gravity's negative work, consistent with the backpack moving at nearly constant speed so its kinetic energy does not change much.

Try The Numbers Yourself

Lift the same backpack through the same height in $2\,\mathrm{s}$ instead of $4\,\mathrm{s}$. The work and energy change stay at $196\,\mathrm{J}$, but the average power changes:

Check that halving the time exactly doubles the power while leaving work untouched. That single comparison is the cleanest way to feel the difference between energy and power.

Calculation Traps To Watch

• Using $W = Fd$ when the force is not along the displacement. With an angle, the correct constant-force form is $W = Fd\cos\theta$.
• Confusing energy with power. Energy answers "how much"; power answers "how fast."
• Forgetting the condition behind a formula. $W = Fd\cos\theta$ is the simple form for a constant force, and $\Delta U_g = mg\Delta h$ is the near-Earth approximation.
• Thinking positive work always means speed increases. It is the net work that determines the change in kinetic energy.
• Treating watts and joules as the same unit. A joule is energy; a watt is joule per second.

A Quick Check Before The Algebra

Ask which quantity the problem really wants:

1. How much energy was transferred by a force? Use work.
2. Stored or changing motion/position energy? Use an energy equation.
3. How quickly the transfer happened? Use power.

These ideas show up whenever forces, motion, and energy transfer matter together: lifting objects, braking, motors, climbing stairs, falling bodies, and machine efficiency. The topic is also the foundation for the work-energy theorem and many conservation-of-energy problems.