Unit 1: Kinematics

Kinematics

Branch of mechanics that describes how objects move (position, velocity, acceleration) without analyzing the forces causing the motion.

Model (in kinematics)

A consistent mathematical description of motion based on a chosen coordinate system, variables, and relationships between them.

Coordinate system

A chosen origin and set of axes/directions used to assign numerical positions to objects; must remain consistent throughout a problem.

Origin

The reference point where position is defined as zero in a chosen coordinate system.

Positive direction convention

The chosen direction labeled as positive; determines the signs of position, displacement, velocity, and acceleration.

Reference frame

The viewpoint from which positions and times are measured (e.g., ground frame, moving train frame).

Inertial reference frame

A non-accelerating frame in which Newton’s laws apply in their standard form; typically assumed in AP Physics C unless stated otherwise.

Position (1D)

Location along a line relative to an origin, often written as a function of time: $x(t)$.

Position vector (2D)

Vector from the origin to the object, written r⃗(t)=x(t)î+y(t)ĵ.

Distance

Total length of the path traveled; always nonnegative and depends on the path taken.

Displacement

Signed change in position from initial to final: Δx = xf − xi (path-independent in 1D/along a line).

Time interval

Elapsed time between two instants: Δt = tf − ti (usually positive if labels are assigned correctly).

Average speed

Total distance traveled divided by elapsed time: (total distance)/Δt.

Average velocity

Displacement divided by elapsed time: $v_{avg} = \frac{\triangle x}{\triangle t}$.

Instantaneous velocity

Velocity at a specific instant; the limit of average velocity as Δt→0, equal to dx/dt.

Derivative (as used in kinematics)

A calculus operation giving an instantaneous rate of change (e.g., velocity is the derivative of position).

Slope of an x–t graph

The instantaneous velocity at that time (positive slope → v>0, negative slope → v<0).

Speed (in 1D)

Magnitude of velocity: |v|; ignores direction.

Turning point

A time when the object momentarily has v=0 and may reverse direction; occurs at a local max/min of x(t).

Average acceleration

Change in velocity over elapsed time: $a_{avg} = \frac{\triangle v}{\triangle t}$.

Instantaneous acceleration

Acceleration at a specific instant; $a(t)=\frac{dv}{dt}$ and also $a(t)=\frac{d^2x}{dt^2}$.

Second derivative of position

$\frac{d^2x}{dt^2}$; equals acceleration and relates to the concavity (curvature) of $x(t)$.

Concavity of an x–t graph

Shape indicating acceleration: concave up → $a > 0$, concave down → $a < 0$.

Constant velocity

Motion with unchanging velocity; implies zero acceleration.

Uniform (constant) acceleration

Motion with acceleration that remains constant over time; enables the standard constant-acceleration equations.

Constant-acceleration equation: v = v0 + at

Relates velocity to time under constant acceleration.

Constant-acceleration equation: $x = x_0 + v_0 t + \frac{1}{2}at^2$

Gives position versus time for constant acceleration in 1D.

Constant-acceleration equation: $v^2 = v_0^2 + 2a(x - x_0)$

Relates velocity and displacement without time for constant acceleration.

Constant-acceleration equation: $\triangle x = \frac{1}{2}(v + v_0)t$

Displacement under constant acceleration using average of initial and final velocity.

“Big Five” kinematic quantities

The five variables linked by constant-acceleration equations in 1D: v, v0, a, Δx, t.

Big Five strategy (missing variable method)

For constant-acceleration problems, pick the equation that does not include the variable neither given nor asked for.

Alternate symbols in kinematics (u, s)

Common substitutions: u for initial velocity v0, and s for displacement Δx.

Free fall

Motion under gravity alone (air resistance neglected), with approximately constant acceleration near Earth’s surface.

Acceleration due to gravity (g)

Magnitude of gravitational acceleration near Earth’s surface, about $9.8 \, m/s^2$ (often approximated as $10 \, m/s^2$).

Sign convention for gravity

If up is positive, a = −g; if down is positive, a = +g (physics unchanged if consistent).

Weightlessness (in free fall)

Condition of zero apparent weight during free fall (support force is zero), even though gravity acts.

Terminal velocity

Constant falling speed reached when drag balances weight, making acceleration zero.

Integral relationship: Δv = ∫ a(t) dt

Change in velocity equals the area under the acceleration–time graph over the interval.

Integral relationship: Δx = ∫ v(t) dt

Displacement equals the signed area under the velocity–time graph over the interval.

Initial condition

A known value (e.g., v(0), x(0)) used to determine the constant of integration after integrating.

Constant of integration

An unknown constant introduced by integration; determined using an initial condition.

Area under a v–t graph

Signed displacement over a time interval (negative if the graph is below the time axis).

Area under an a–t graph

Change in velocity over a time interval.

Vector quantity

A quantity with magnitude and direction (e.g., displacement, velocity, acceleration).

Unit vector

A vector of magnitude 1 used to indicate direction (e.g., î and ĵ).

Components (of a vector)

Perpendicular parts of a vector along coordinate axes (e.g., vx and vy) that can be analyzed independently in many problems.

Speed from components

Magnitude of velocity in 2D: $|\vec{v}| = \sqrt{v_x^2 + v_y^2}$.

Projectile motion (ideal)

2D motion with only gravitational acceleration: $a_x=0$ and $a_y=-g$ (no air resistance).

Galilean relativity (velocity addition)

In non-relativistic frames, time is the same in all frames and velocities add linearly: $\vec{v} = \vec{v}' + \vec{u}$ (or $\vec{v}'=\vec{v}-\vec{u}$).

Piecewise motion model

A motion description split into stages where rules (like acceleration) differ; stages are linked by boundary conditions so x and usually v are continuous at transition times.