AP Physics 1: Work, Energy, and Power

Work, energy, and power form one of the most useful toolkits in AP Physics 1 because they let you solve motion problems without tracking every force at every instant. Instead of focusing on acceleration at each moment, energy methods connect what happens at the start and end of a process. Used well, they simplify everything from block-and-ramp questions to springs, pulleys, and collisions with external forces.

Work and the Work-Energy Theorem

What “work” means in mechanics

In physics, work is not “effort.” It is the transfer of energy by a force acting through a displacement. For a constant force applied at an angle $\theta$ to the displacement, the work is

$W=Fd\cos \theta$

Key implications:

• Only the component of the force parallel to the displacement contributes.
• If force is perpendicular to displacement, $\cos 90^{\circ }=0$, so the work is zero. (A classic example is uniform circular motion: centripetal force does no work because it is perpendicular to the motion.)

Work is measured in joules (J), where $1\text{ J}=1\text{ N}\cdot \text{m}$.

The work-energy theorem

The central result is the work-energy theorem:

$$W_{\text{net}}=\Delta K$$

where $W_{\text{net}}$ is the net work done on the object, and $K$ is kinetic energy. Kinetic energy is

$K=\frac{1}{2}m{v}^2$

So if you can compute net work, you can directly connect it to a change in speed without solving for time or acceleration.

A practical example: if a block slides on a rough surface and friction does negative work, you can find how much its kinetic energy decreases over a given distance. That is often simpler than using $a=F_{\text{net}}/m$ and kinematics.

Forms of Mechanical Energy

Kinetic energy

Kinetic energy depends on speed, not direction. Doubling speed quadruples kinetic energy, which matters in questions about braking distance, impact energy, or how much work an engine must do to increase speed.

Potential energy: gravitational and spring

Potential energy describes stored energy associated with position or configuration.

Gravitational potential energy near Earth’s surface is

$U_g=mgh$

Only changes in height matter, so $\Delta U_g=mg\Delta h$. This is why many ramp problems ignore ramp length and focus on vertical rise.

Elastic potential energy stored in a spring is

$U_s=\frac{1}{2}k{x}^2$

where $k$ is the spring constant and $x$ is displacement from equilibrium. The square dependence means compressing a spring twice as far stores four times the energy.

Conservation of Mechanical Energy and When It Applies

The conservation idea

If only conservative forces do work, the total mechanical energy stays constant:

$$K_i+U_i=K_f+U_f$$

Conservative forces are those whose work depends only on initial and final positions, not the path taken. Gravity and spring forces are conservative in AP Physics 1 contexts.

This approach is powerful because it skips forces entirely. For example, a cart released from rest at height $h$ converts gravitational potential energy into kinetic energy. If friction and air resistance are negligible, then

$mgh=\frac{1}{2}m{v}^2$

and mass cancels, making the final speed independent of mass.

What to do when nonconservative forces are present

Friction, air resistance, and applied pushes or pulls often do work that changes mechanical energy. In those cases, use

$$K_i+U_i+W_{\text{nc}}=K_f+U_f$$

where $W_{\text{nc}}$ is work by nonconservative forces. Kinetic friction typically does negative work:

$W_f=-f_{k}d$

and since $f_k={\mu }_{k}N$, you can connect energy loss to distance traveled and surface properties.

This is also where energy methods shine: instead of tracking changing acceleration on a rough incline, you can treat friction’s work as an energy “cost” that reduces the final kinetic energy.

Choosing Energy Methods vs. Force Methods

A common AP Physics 1 skill is deciding whether to use an energy approach or a force approach.

Use energy when:

• You care about speeds or heights between two points, not the time.
• Forces are complicated but the net energy change is easy to express.
• Motion involves springs, vertical changes, or friction over a distance.
• The problem is “start to finish” and does not demand acceleration at a specific point.

Use forces when:

• You need acceleration, tension, normal force, or friction force itself.
• The problem asks about dynamics at an instant (for example, maximum tension at the bottom of a swing).
• Constraints create forces you must compute (like contact forces in circular motion).

A good strategy is to start with energy for the big picture, then use Newton’s second law only if the question asks for a force.

Power: The Rate of Energy Transfer

Work and energy tell you “how much.” Power tells you “how fast.”

Average power is

$P_{\text{avg}}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}$

Instantaneous mechanical power for a force applied to an object moving with velocity $\vec{v}$ is

$P=\vec{F}\cdot \vec{v}=Fv\cos \theta$

Power is measured in watts (W), where $1\text{ W}=1\text{ J/s}$.

This shows up in real contexts: engines, elevators, and athletes. For example, lifting a mass to a given height requires work $mgh$ regardless of how slowly you lift it. But lifting it faster requires greater power because the same energy transfer occurs in less time.

Mechanical Advantage and Efficiency

Energy methods also clarify how machines help us.

Mechanical advantage: trading force for distance

In ideal machines, you do not get “free energy.” You trade force for distance. A simple idea is that input work approximately equals output work:

$W_{\text{in}}\approx W_{\text{out}}$

If a ramp lets you raise a box by height $h$ using a smaller force over a longer distance $d$, the energy requirement stays about the same:

$F_{\text{in}}d\approx mgh$

This explains why longer ramps feel easier: they reduce the needed force, not the required energy.

Efficiency: accounting for losses

Real machines have friction and other losses, so input work exceeds useful output work. Efficiency is

$$\eta =\frac{W_{\text{useful out}}}{W_{\text{in}}}=\frac{P_{\text{useful out}}}{P_{\text{in}}}$$

Efficiency is often expressed as a percentage. If a pulley system lifts a load but heats up due to friction, that thermal energy is part of the input but not useful output, lowering efficiency.

In AP Physics 1, efficiency problems often ask you to compute the extra input work required given an efficiency. If $\eta =0.80$, then

$W_{\text{in}}=\frac{W_{\text{useful out}}}{0.80}$

Common AP Physics 1 Pitfalls

• Sign errors with work: friction does negative work when it opposes motion; gravity can do positive or negative work depending on direction of displacement.
• Confusing potential energy with force: $U_g=mgh$ is not a force. The force is weight $mg$.
• Forgetting the system choice: conservation of energy works cleanly when you define the system so that internal conservative forces are handled through potential energy.
• Using mechanical energy conservation when it does not apply: if an external push or friction is significant, include $W_{\text{nc}}$ rather than forcing $K+U$ to be constant.

Bringing It Together

Work, energy, and power unify a wide range of mechanics problems under a few consistent principles. The work-energy theorem links net work to changes in speed. Potential energy accounts for gravity and springs without tracking forces step-by-step. Conservation of energy gives a start-to-finish shortcut when nonconservative effects are absent, and the extended energy equation handles friction and external work when they are present. Power adds the time dimension, and mechanical advantage and efficiency connect physics to real machines. Mastering these ideas is less about memorizing formulas and more about recognizing which energy changes matter and which forces add or remove energy from the system.