AP Physics C: Mechanics · Unit 3: Work, Energy, and Power · Lesson 3.5

Deep Dive: Power

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

3.5.A.1Concept

What Is Power?

Power is the rate at which energy changes with respect to time — whether that's energy transferring into or out of a system, or energy converting from one form to another within it. Everything from Lessons 3.1 through 3.4 described how much energy moved. Power describes how fast.

💡Two identical cars can do the exact same amount of work accelerating to highway speed — but if one does it in 5 seconds and the other in 15, they deliver wildly different power, even though the total energy transferred is identical.

3.5.A.23.5.A.3Math

Average Power

Average power is simply the energy transferred or converted, divided by the time that transfer took:

P(avg) = ΔE / Δt

Since work is itself the change in energy due to a force (Lesson 3.2), average power has an equally direct form in terms of work:

P(avg) = ΔW / Δt

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Instantaneous Power

Just like every other "instantaneous" quantity in this course, instantaneous power is a derivative — this time, of work with respect to time:

P(inst) = dW/dt

For a constant force, this simplifies beautifully — recall the dot product from Lesson 3.2, now applied between force and velocity instead of force and displacement:

P(inst) = F⃗ · v⃗ = Fv cos θ

🔑Same structure as work, same intuition: only the component of force parallel to velocity contributes to power. A force entirely perpendicular to motion — like the centripetal force in uniform circular motion — delivers exactly zero power, no matter how large it is.

Explore this directly below, using the generator from this unit's header art.

The generator from this unit's header art. Set the applied force, the object's speed, and the angle between them — watch the power gauge respond, including swinging negative when the force opposes motion (like braking).

F (N)500

v (m/s)12

θ0°

P = (500)(12)cos(0°) = 6000 W

Positive power — the force is doing work that adds energy to the object's motion.

ExampleGuided Example — Power Delivered While Braking

A 1500 kg car moving at 20 m/s applies its brakes, which exert a 6000 N force directly opposite its velocity. Find the instantaneous power delivered by the braking force at this moment.

Step 1 — Identify the angle between force and velocity

The braking force points directly opposite the car's motion, so θ = 180°.

Math

Applying Power: From P(t) to v(t)

A genuinely useful application: if an engine delivers constant powerto a car starting from rest, you can derive the car's entire velocity function using nothing but the definitions from this lesson.

ExampleWorked Example — Deriving v(t) from Constant Power

A car of mass m starts from rest, and its engine delivers constant power P to the wheels (ignore friction and air resistance). Derive v(t).

This is exactly the relationship you can explore live below.

Exactly the scenario the CED itself suggests: a car's engine delivers constant power P from rest on a frictionless road. Starting from P = Fv = m(dv/dt)v and separating variables gives v(t) = √(2Pt/m) — velocity keeps climbing, but ever more slowly, since the available force (F = P/v) shrinks as speed increases.

Power (W)60 kW

Mass (kg)1200

time (s)5.0 s

v(t) = 22.36 m/sa(t) = P/(mv) = 2.24 m/s²

Notice acceleration keeps shrinking over time even though power stays perfectly constant — this is exactly why a real car "runs out of punch" at highway speed even at full throttle.

💡Notice v(t) grows like √t, not linearly — which means acceleration (the derivative of this function) actually decreases over time, even under perfectly constant power. This is a genuinely useful, testable AP-level result, and it's exactly the scenario the course's own suggested activities point to.

← Back to Lesson 3.5That's a wrap on Unit 3! Unit 4 shifts focus to momentum — impulse, collisions, and a second conservation law.