3.5 Power

Power is how fast energy moves or changes form, measured in watts, or joules per second. Use two main ideas: average power, which is energy change divided by time, and instantaneous power, which connects force and velocity at a single moment.

Why This Matters for the AP Physics 1 Exam

Power ties together everything from earlier in Unit 3: kinetic energy, work, potential energy, and conservation of energy. On the exam, you might calculate how quickly a motor, person, or engine transfers energy, or explain energy flow in a system using power language.

The first free-response question, the Mathematical Routines question, asks you to create and use mathematical models, calculate or derive an expression, and write a clear, reasoned explanation. Power problems are good practice for this because they often combine a calculation with a short justification about energy transfer. Power can also show up in multiple-choice questions that test whether you can connect force, velocity, work, and time without getting lost in the units.

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Key Takeaways

• Power is the rate of energy change, whether energy moves into a system, out of it, or converts from one form to another inside it.
• Average power is energy change over time: $P_{avg} = \frac{\Delta E}{\Delta t}$, and since work changes energy, $P_{avg} = \frac{W}{\Delta t}$.
• Instantaneous power for a constant force is $P_{inst} = F_{\|} v = Fv\cos\theta$, using only the force component parallel to velocity.
• The SI unit of power is the watt (W), where 1 W = 1 J/s.
• The sign of power tells direction: positive adds energy to the system, negative removes it, and zero happens when force is perpendicular to velocity.
• When an object moves at constant speed, net power is zero, so any power input is balanced by power lost to opposing forces.

Energy Transfer and Power

Power as Energy Rate

Power measures how quickly energy changes or transfers over time. It represents the rate at which energy flows into or out of a system, or converts from one form to another within a system.

• Energy can be transferred into or out of a system at different rates
• Within a system, energy can be converted between forms (like potential to kinetic)
• Power is measured in watts (W), where 1 watt equals 1 joule per second (J/s)

Average Power Calculation

Average power tells you the rate of energy transfer over a period of time.

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

This equation shows that average power equals the change in energy divided by the time interval during which that change occurs. A few applications that show the idea:

• A 60-watt lightbulb transfers 60 joules of electrical energy into light and heat every second
• A 1500-watt hair dryer running for 5 minutes transfers 450,000 joules of energy (1500 W × 300 s)
• A solar panel producing 300 watts generates 300 joules of electrical energy each second from sunlight

Power as Work Rate

When a force causes displacement, you can think of power as how quickly work is being done.

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

Since work is a form of energy transfer, power can be calculated by dividing the work done by the time taken:

• A crane lifting a heavy load does work against gravity at a certain power
• A person pushing a cart across a floor does work against friction, with power depending on speed and force
• An engine performs work to accelerate a vehicle, with its power rating showing how quickly it can do this work

Instantaneous Power from Force

Instantaneous power gives the exact rate of energy transfer at a specific moment, which is useful for analyzing systems with varying forces or velocities.

$P_{inst} = F_{||} v = Fv \cos \theta$

This equation relates instantaneous power to force and velocity:

• $F_{||}$ is the component of force parallel to the velocity
• $v$ is the object's velocity
• $\theta$ is the angle between the force and velocity vectors

The sign of power indicates the direction of energy transfer:

• Positive power means energy is being added to the system (like a car engine accelerating the vehicle)
• Negative power indicates energy is being removed from the system (like brakes slowing a car)
• Zero power occurs when force and velocity are perpendicular (like centripetal force in circular motion)

How to Use This on the AP Physics 1 Exam

Problem Solving

Start by deciding which equation fits the question. If you know an energy change or work done over a time interval, use $P_{avg} = \frac{\Delta E}{\Delta t}$ or $P_{avg} = \frac{W}{\Delta t}$. If you know a force and a velocity at one instant, use $P_{inst} = Fv\cos\theta$.

A reliable approach:

1. Identify whether the question wants average or instantaneous power.
2. Find the relevant force, energy change, or work.
3. Watch the angle between force and velocity. Only the parallel component does work.
4. Check your units. Power should come out in watts (J/s).

Free Response

The Mathematical Routines question rewards clear reasoning, not just a final number. When you explain a power result, say where the energy comes from and where it goes. For example, in a constant-speed situation, state that net work is zero, so the power input is fully balanced by power lost to opposing forces. A coherent, step-by-step explanation that cites a physical principle earns more than a bare calculation.

Common Trap

When an object moves at constant speed, students sometimes think the engine does no useful work because kinetic energy is not changing. The engine still delivers power; that power just goes into overcoming friction and air resistance instead of speeding the object up.

Practice Problem 1: Average Power Calculation

Solution

To solve this, find the work done against gravity and then calculate the average power:

1. Calculate the gravitational force on the student: $F = mg = 60 \text{ kg} \times 9.8 \text{ m/s²} = 588 \text{ N}$

2. Calculate the work done in climbing the stairs: $W = F \times h = 588 \text{ N} \times 5 \text{ m} = 2940 \text{ J}$

3. Calculate the average power: $P_{avg} = \frac{W}{\Delta t} = \frac{2940 \text{ J}}{8 \text{ s}} = 367.5 \text{ W}$

The student's average power output is 367.5 watts.

Practice Problem 2: Instantaneous Power from Force

Solution

To find the instantaneous power output of the engine, use the equation for instantaneous power:

$P_{inst} = F_{||} v = Fv \cos \theta$

Since the engine force is in the same direction as the velocity (θ = 0°): $P_{inst} = 600 \text{ N} \times 20 \text{ m/s} \times \cos(0°) = 600 \text{ N} \times 20 \text{ m/s} = 12,000 \text{ W} = 12 \text{ kW}$

Energy transfers in this situation:

1. The engine converts chemical energy from fuel to mechanical energy at a rate of 12 kW
2. This energy is transferred to overcome the opposing forces of air resistance and friction
3. Since the car moves at constant speed, the net work done on the car is zero (forces are balanced)
4. All of the engine's power output (12 kW) is being converted to heat and sound through friction and air resistance

Practice Problem 3: Power Supplied by a Motor

Solution

When the elevator moves at constant speed, the motor must supply an upward force equal to the elevator's weight:

1. Calculate the force: $F = mg = 2000 \text{ kg} \times 9.8 \text{ m/s²} = 19{,}600 \text{ N}$

2. Calculate the instantaneous power: $P_{\text{inst}} = F_{||}v = 19{,}600 \text{ N} \times 3 \text{ m/s} = 58{,}800 \text{ W} = 58.8 \text{ kW}$

The motor supplies 58.8 kW of power.

Common Misconceptions

• Power and energy are not the same. Energy (in joules) is the total amount transferred; power (in watts) is how fast that transfer happens. A small motor can do the same work as a large one if you give it more time.
• More force does not always mean more power. Instantaneous power depends on both force and velocity. A large force on a stationary object delivers zero power because v = 0.
• A perpendicular force delivers no power. When force and velocity are at 90 degrees, $\cos\theta = 0$, so the power is zero even though a force is present, as with centripetal force in circular motion.
• Constant speed does not mean zero power input. Net work is zero, but the engine or motor still supplies power that is balanced by power lost to friction and air resistance.
• Negative power is real and meaningful. It does not mean a mistake. It signals that energy is leaving the system, like when brakes slow a car.

Related AP Physics 1 Guides

• 3.2 Work
• 3.3 Potential Energy
• 3.4 Conservation of Energy
• 3.1 Translational Kinetic Energy