Kinematics: every key term you need (+ practice quiz)

More AP Physics C guides

• Newton's Laws of Motion
• Work, Energy and Power
• Systems of Particles and Linear Momentum
• Rotational Motion
• Gravitation

Position

x(t); location relative to an origin along an axis; a vector in 1D given by sign; the starting point for all kinematics.

Velocity as a Derivative

v = dx/dt; the instantaneous rate of change of position; slope of the position-time graph at a point.

Acceleration as a Derivative

a = dv/dt = d²x/dt²; instantaneous rate of change of velocity; slope of the velocity-time graph.

Average vs Instantaneous

Average velocity = Δx/Δt (secant slope); instantaneous velocity = dx/dt (tangent slope) as Δt→0.

Position from Velocity (Integral)

x(t) = x₀ + ∫v dt; displacement is the area under the velocity-time graph.

Velocity from Acceleration (Integral)

v(t) = v₀ + ∫a dt; change in velocity is the area under the acceleration-time graph.

Kinematic Equation 1

v = v₀ + at; valid only for constant acceleration; follows from integrating a = constant once.

Kinematic Equation 2

x = x₀ + v₀t + ½at²; constant acceleration; follows from integrating v = v₀ + at.

Kinematic Equation 3

v² = v₀² + 2a(x − x₀); constant acceleration, time-independent; derived by eliminating t.

Free Fall

Constant downward acceleration g ≈ 9.8 m/s² near Earth's surface; independent of mass when air resistance is neglected.

Projectile Motion

Independent horizontal (aₓ = 0) and vertical (a_y = −g) motions sharing one time variable t.

Horizontal Projectile Component

x = v₀ₓt with constant vₓ = v₀cosθ; horizontal velocity is unchanged throughout the flight.

Vertical Projectile Component

y = v₀_y t − ½gt² with v_y = v₀sinθ − gt; vertical motion is free fall.

Trajectory Shape

Eliminating t from the projectile equations gives y as a quadratic in x — a parabola.

Range of a Projectile

For launch and landing at the same height, R = v₀²sin(2θ)/g; maximum at θ = 45°.

Time of Flight

For level ground, t = 2v₀sinθ/g; found by setting the vertical displacement back to zero.

Relative Velocity

v_AC = v_AB + v_BC; velocities add as vectors when changing reference frames.

Position-Time Graph

Slope gives velocity; a curve means changing velocity (acceleration); concavity shows the sign of a.

Velocity-Time Graph

Slope gives acceleration; the signed area between the curve and the axis gives displacement.

Sign Conventions

Choose a positive direction first; velocity and acceleration can have opposite signs (object slowing down).

Turning Point

Where velocity is momentarily zero and reverses sign; acceleration need not be zero there (e.g. top of a toss).

Non-Constant Acceleration

When a depends on t, integrate directly: v(t) = v₀ + ∫₀ᵗ a(t′)dt′; the constant-a kinematic equations do not apply.

Velocity-Dependent Acceleration

If a = f(v), separate variables: ∫dv/f(v) = ∫dt; e.g. a = −kv gives v = v₀e^(−kt).

Position-Dependent Acceleration

If a = f(x), use a = v dv/dx so ∫v dv = ∫f(x)dx; this is the chain rule dv/dt = (dv/dx)(dx/dt).

Terminal Velocity

With drag a = g − (b/m)v, acceleration vanishes when v = mg/b; velocity approaches this value asymptotically.

Chain Rule Identity

a = v(dv/dx) = ½ d(v²)/dx; useful whenever acceleration is given as a function of position.

Vector Kinematics

r(t) = x(t)î + y(t)ĵ; v = dr/dt and a = dv/dt are found component-by-component.

Speed vs Velocity Magnitude

Speed = |v| = √(vₓ² + v_y²); average speed = distance/time, which can exceed |average velocity|.

Projectile Apex Condition

At maximum height v_y = 0 while vₓ = v₀cosθ ≠ 0, so speed at the apex is v₀cosθ, not zero.

Projectile onto Incline

Set the trajectory y(x) equal to the incline line y = x tanφ and solve for x to find where it lands.

Complementary Launch Angles

Angles θ and 90° − θ give the same range on level ground because sin(2θ) = sin(180° − 2θ).

Maximum Height

H = v₀²sin²θ/(2g); found from v_y² = v₀_y² − 2gH with v_y = 0.

Relative Motion in 2D

Positions and velocities transform by vector subtraction: v_A/B = v_A − v_B; accelerations are the same in all inertial frames.

Curvature of x-t Graph

Concave up means a > 0; concave down means a < 0; an inflection point marks a = 0 momentarily.

Graphical Integration Sign

Area below the t-axis on a v-t graph counts as negative displacement; total distance uses absolute areas.

Exponential Approach

Solutions like v = v_T(1 − e^(−t/τ)) have a time constant τ; after one τ the quantity is 63% of the way to its limit.