Kinematics

What this unit covers

The topics below follow the published Physics C: Mech course framework for Unit 1. This unit is worth 10–15% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Motion as Derivatives13 min · 3 objectivesDefine velocity and acceleration as the first and second derivatives of position · Differentiate a polynomial position function x(t) to find v(t) and a(t) · Interpret the signs of v and a to decide whether an object speeds up or slows down
• Integrating Acceleration14 min · 3 objectivesRecover velocity from acceleration and position from velocity by integration · Use initial conditions to fix the constant of integration · Derive the constant-acceleration kinematic equations as integrals of a = constant
• Vectors & Projectile Motion14 min · 3 objectivesResolve a vector into perpendicular components and combine components back into a magnitude · Treat 2D motion as two independent 1D problems sharing a clock · Analyze projectile motion using aₓ = 0 and a vertical acceleration of −g
• Relative Motion12 min · 3 objectivesExpress velocities relative to different reference frames and convert between them · Apply the relative-velocity addition rule in one and two dimensions · Solve river-crossing and crosswind problems using vector components

Formulas in Unit 1

Every term in Unit 1

All 18 terms we publish for Kinematics, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Velocity as a derivative

v = dx/dt. The constant-acceleration equations are a special case; when a varies, differentiate and integrate instead.

Acceleration as a derivative

a = dv/dt = d²x/dt². Also a = v(dv/dx), which is the form to use when acceleration is given as a function of position.

Integrating to find position

x = ∫v dt and v = ∫a dt, with the constant of integration fixed by the initial condition. Forgetting that constant is the standard error.

Area under a velocity-time curve

The definite integral ∫v dt gives displacement. Area below the axis subtracts.

Non-constant acceleration

When a = a(t), integrate directly. When a = a(v) or a = a(x), separate variables before integrating.

Projectile motion with calculus

Independent components: x = v₀cos θ·t and y = v₀sin θ·t − ½gt². Solve the vertical equation for time and substitute.

Relative velocity

v_AC = v_AB + v_BC as vectors. Reference frames moving at constant velocity are equally valid for Newtonian mechanics.

Deriving v² = v₀² + 2aΔx

From a = v dv/dx, separate and integrate ∫v dv = ∫a dx. Shows why this relation contains no time variable.

Deriving an expression symbolically

Answer in terms of the given symbols before substituting numbers. Marks are awarded for the expression, and a numerical slip then costs only one point.

Checking a limiting case

Test your expression as a variable goes to zero or infinity. If a = g sin θ/(1 + I/MR²) does not reduce to g sin θ when I = 0, the derivation is wrong.

Dimensional analysis as a check

Every term in a sum must have the same units. A quick check catches most algebraic errors before they propagate.

Graph of a(t) to v(t)

Integrate: the area under the acceleration-time curve is the change in velocity, and the initial value fixes the constant.

Concavity of a position graph

Concave up means positive acceleration, concave down negative. An inflection point is where acceleration changes sign.

When kinematic equations do not apply

Any time acceleration is not constant — including drag, springs and gravitation at astronomical scales. Use calculus or energy instead.

Average vs instantaneous quantities

Average velocity is Δx/Δt; instantaneous is dx/dt. They coincide only for constant velocity.

Choosing coordinates

Align an axis with the acceleration where possible. On an incline, tilting the axes eliminates one component of the normal force from the equations.

Motion with a = a(v)

Separate variables: m dv/a(v) = dt. This is the standard route for drag problems.

Motion with a = a(x)

Use a = v dv/dx and separate: v dv = a(x) dx. Integrating gives a velocity-position relation without solving for time.

What examiners penalize here

• On the AP exam, given x(t) you should be able to produce v(t) and a(t) instantly by differentiating, and read the reverse too: where the x–t slope is zero the object is momentarily at rest, and where the v–t slope is zero the acceleration is zero.
• Keep the subscript bookkeeping strict: write every velocity as v(object/frame). The chain v(A/C) = v(A/B) + v(B/C) only works when the inner labels match, and swapping any pair introduces a sign flip. Getting the subscripts right turns most relative-motion questions into simple addition.

Practice Physics C: Mech

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 7 units of AP Physics C: Mechanics

1. Unit 1 · Kinematics
2. Unit 2 · Force and Translational Dynamics
3. Unit 3 · Work, Energy, and Power
4. Unit 4 · Linear Momentum
5. Unit 5 · Torque and Rotational Dynamics
6. Unit 6 · Energy and Momentum of Rotating Systems
7. Unit 7 · Oscillations