AP Physics C: Mechanics · Unit 1: Kinematics · Lesson 1.2

Deep Dive: Displacement, Velocity, and Acceleration

🔬 Deep Dive

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

1.2.A.1Concept

The Object Model

Before you can talk about an object's motion, you need to decide how much detail actually matters. The object model throws out size, shape, and internal structure entirely and treats the object as a single point — carrying only extensive properties like mass (and, later in the course, charge).

💡This isn't a simplification you'll ever "graduate" from — it's the working assumption behind essentially every kinematics problem in this course. A car, a planet, a proton: as far as kinematics is concerned, they're all just points.

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Displacement

Displacement is the change in an object's position — where it ended up, minus where it started. It says nothing about the path taken in between.

Δx = x − x₀

🔑Displacement cares only about the start and end points. A runner who does a full lap around a track has a displacement of zero — they end up exactly where they started — even though the distance they covered was very much not zero.

1.2.B.11.2.B.21.2.B.31.2.B.4ConceptMath

Average Velocity and Average Acceleration

Averages of velocity and acceleration are calculated from the initial and final states of an object over some interval of time — you don't need to know anything about what happened in between.

vavg = Δx / Δt

aavg = Δv / Δt

⚠️An object is accelerating whenever the magnitude ordirection of its velocity changes — not just when it's speeding up. A car taking a turn at constant speed is accelerating, because its direction is changing even though its speed isn't.

ExampleWorked Example — Average Velocity and Acceleration

A drone is at x = 2 m and moving at 3 m/s at t = 0 s. At t = 5 s, it's at x = 22 m, moving at 11 m/s. Find its average velocity and average acceleration over this interval.

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Approaching the Instant

Here's the key idea that unlocks calculus-based kinematics: calculating an average velocity or average acceleration over a very small time interval gives a value that's very close to the instantaneous velocity or acceleration at that moment. The smaller the interval, the closer the approximation.

🔑This isn't just a computational trick — it's the entire idea behind the derivative. "Instantaneous" always means "the limit of an average, as the interval shrinks to zero." Keep that phrase in mind; it'll define nearly every "instantaneous" quantity for the rest of the course.

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Instantaneous Values as Derivatives

As the time interval used to calculate an average shrinks toward zero, that average approaches the value of the quantity at a single instant — the instantaneous value. This limiting process is exactly what a derivative is.

Instantaneous velocity

Instantaneous velocity is the rate of change of position with respect to time — the derivative of position:

v⃗ = dr⃗ / dt

vₓ = dx / dt

Instantaneous acceleration

Instantaneous acceleration is the rate of change of velocity with respect to time — the derivative of velocity:

a⃗ = dv⃗ / dt

aₓ = dvₓ / dt

Use the tool below to see this limiting process directly: watch the average-velocity secant line rotate into the instantaneous-velocity tangent line as Δt shrinks toward zero.

The curve is x(t) = 2t² + 1. The reference point is fixed at t₁ = 1.5 s. Shrink Δt and watch the secant line (average velocity) rotate toward the tangent line (instantaneous velocity).

Δt1.50 s

vavg = Δx/Δt = 9.00 m/svₓ at t₁ (true derivative) = 6.00 m/s

Drag Δt all the way down — the violet secant slope converges on the green tangent slope. That limit is the derivative dx/dt.

💡Graphically: instantaneous velocity is the slope of the line tangentto a position-vs-time graph at a point. Instantaneous acceleration is the slope of the tangent line on a velocity-vs-time graph. This is the same "rise over run" idea from average velocity — just zoomed all the way in to a single point.

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Differentiation and Integration Across Position, Velocity, and Acceleration

Position, velocity, and acceleration are all connected by derivatives — which means they're also connected by integrals, since integration undoes differentiation. This gives you two directions to move in:

Differentiate ⟶

x(t) → v(t) → a(t). Take the derivative to go from position to velocity, and again to go from velocity to acceleration.

⟵ Integrate

a(t) → v(t) → x(t). Integrate to go backward — from acceleration to velocity, and from velocity to position (plus a constant of integration, fixed by an initial condition).

Explore the connection yourself below. All three graphs describe the samemotion — dragging the time slider moves a synchronized point on each one.

Same motion, three graphs. Drag the slider — the dot on each graph moves together, showing how differentiating position gives velocity, and differentiating velocity gives acceleration.

time t1.50 s

x(t) = 2t² + 1 (position)

v(t) = dx/dt = 4t (velocity)

a(t) = dv/dt = 4 (acceleration)

x = 5.50 mv = 6.00 m/sa = 4.00 m/s²

ExampleGuided Example — Differentiating a Position Function

An object's position is given by x(t) = 3t² − 2t + 5 (meters, t in seconds). Find its velocity and acceleration as functions of time, and evaluate both at t = 2 s.

Step 1 — Differentiate position to get velocity

v(t) = dx/dt = 6t − 2 (m/s)

ExampleWorked Example — Integrating an Acceleration Function

An object starts at rest (v₀ = 0 m/s) with acceleration a(t) = 6t (m/s²). Find v(t), then find the object's displacement between t = 0 s and t = 3 s.

← Back to Lesson 1.2Ready for 1.3? Representing Motion connects everything here to graphs — and adds the constant-acceleration kinematic equations.