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
• Motion Graphs: Reading Derivatives and Areas14 min · 3 objectivesMove between position, velocity and acceleration graphs using slopes and areas · Extract displacement from a velocity graph by computing signed area · Distinguish displacement from distance traveled on a graph that changes sign
• Non-Constant Acceleration & Separation of Variables14 min · 3 objectivesRecognize when the constant-acceleration kinematic equations do not apply · Solve motion problems in which acceleration depends on time, velocity or position · Use separation of variables to integrate a velocity-dependent acceleration
• Uniform Circular Motion & Centripetal Acceleration14 min · 3 objectivesExplain why an object in uniform circular motion is accelerating despite constant speed · Relate period, frequency, linear speed and angular speed for circular motion · Compute centripetal acceleration and identify its direction

Formulas in Unit 1

Every term in Unit 1

All 40 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.

Position, displacement, distance and speed

Position is a coordinate; displacement is the change in it, a vector; distance is the path length, a scalar that never decreases; speed is |v| and average speed is distance over time, which is NOT the magnitude of average velocity unless the motion never reverses.

Sign conventions in one dimension

Choose a positive direction once and keep it. A negative velocity means motion in the negative direction; a negative acceleration does not mean slowing down. Most one-dimensional errors are sign errors introduced by changing the convention midway.

Finding turning points on a v–t graph

A turning point is where v crosses zero and changes sign. It is where displacement is locally maximum or minimum, and it is not where the graph is steepest — steepness is acceleration.

Reading acceleration from curvature of x(t)

Concave up means positive acceleration, concave down negative. The slope gives velocity and the curvature gives acceleration, so a single x(t) curve encodes all three quantities.

Free fall as constant acceleration

With air resistance neglected, every object has a = −g = −9.8 m/s² regardless of mass, including at the top of its flight where v = 0 but a is unchanged. "Zero velocity implies zero acceleration" is the classic error.

Independence of horizontal and vertical motion

Projectile motion is two separate one-dimensional problems sharing only the time. Horizontally a = 0 so x = v₀ₓt; vertically a = −g. A bullet fired horizontally and one dropped from the same height land together.

Time of flight on level ground

t = 2v₀ sin θ/g. It is set entirely by the vertical component, so a steeper launch stays airborne longer at the same speed.

Range on level ground

R = v₀² sin(2θ)/g. Maximum at θ = 45°, and proportional to the square of the launch speed — doubling the speed quadruples the range.

Maximum height of a projectile

H = (v₀ sin θ)²/2g, reached when the vertical velocity component is zero. The horizontal component is unchanged there, so the speed at the apex is v₀ cos θ, not zero.

Complementary launch angles

θ and (90° − θ) give the same range on level ground, because sin(2θ) = sin(180° − 2θ). The steeper one takes longer and goes higher; the shallower is flatter and faster.

Trajectory equation

Eliminating t gives y = x tan θ − gx²/(2v₀² cos²θ), a parabola. Useful when a question relates height to horizontal position without mentioning time.

Launching from a height

The symmetric formulas fail: the flight is longer than 2v₀ sin θ/g and the optimum angle for range is less than 45°. Solve the vertical quadratic for t rather than assuming symmetry.

Velocity components during flight

vₓ is constant throughout; v_y decreases linearly through zero at the apex and becomes negative. Speed is minimum at the apex and equal at equal heights on the way up and down.

Speed at a given height

Energy conservation gives v = √(v₀² − 2gh) independent of launch angle. A projectile passes any given height at the same speed regardless of direction of travel.

Galilean velocity addition

v_AC = v_AB + v_BC — velocity of A relative to C is the vector sum. Subscript bookkeeping, with adjacent inner subscripts canceling, prevents nearly every relative-motion error.

River crossing problem

The boat velocity relative to water adds vectorially to the current. Heading straight across gives the shortest time but downstream drift; heading upstream at an angle gives zero drift but a longer crossing.

Minimum time versus minimum drift

Minimum crossing time comes from pointing the bow perpendicular to the bank, since only the perpendicular component shortens the trip. Zero drift requires an upstream component that cancels the current, which reduces the perpendicular speed.

Period and frequency in circular motion

T is the time for one revolution and f = 1/T. Linear speed is v = 2πr/T and angular speed ω = 2π/T, so v = ωr connects them.

Centripetal acceleration

a_c = v²/r = ω²r, always directed toward the center and perpendicular to the velocity. It changes the direction of the velocity without changing its magnitude.

Tangential versus centripetal acceleration

The centripetal component changes direction; the tangential component dv/dt changes speed. They are perpendicular, so the total magnitude is √(a_c² + a_t²) and uniform circular motion has a_t = 0.

Angular position, velocity and acceleration

θ in radians, ω = dθ/dt, α = dω/dt. They stand in the same derivative relationship as their linear counterparts, which is why the rotational kinematic equations look identical.

Arc length and angular displacement

s = rθ, v = rω and a_t = rα, all requiring θ in radians. Forgetting to convert from degrees is the most common numerical error in rotational kinematics.

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.
• Read the question for **displacement** versus **distance** before computing anything. If any part of the velocity graph lies below the axis the two answers differ, and the difference is usually the entire point of the item.

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