Unit 1: Kinematics

Kinematics

The study of motion without analyzing the causes (forces) of the motion.

Reference frame

A chosen coordinate system (origin and positive direction) plus a clock used to measure motion; must be kept consistent.

Position (x)

An object’s location relative to a chosen origin, usually represented by a coordinate such as x in 1D.

Origin

The zero point of a coordinate system from which positions are measured.

Displacement (Δx)

The change in position from initial to final: $\Delta x = x_f - x_i;$ a vector (direction/sign matters).

Distance

The total length of the path traveled; a scalar and never negative.

Scalar

A quantity with magnitude only (e.g., time, speed, distance, energy).

Vector

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

Sign convention

The assignment of which direction is positive/negative in 1D; determines the signs of displacement, velocity, and acceleration.

Time interval (Δt)

Elapsed time: $\Delta t = t_f - t_i;$ typically positive in AP problems.

Average speed

Total distance traveled divided by elapsed time: $\text{speed}_{\text{avg}} = \frac{\text{distance}}{\text{Δt}}$ (nonnegative scalar).

Average velocity (v_avg)

Displacement divided by elapsed time: $v_{\text{avg}} = \frac{\Delta x}{\Delta t}$ (vector; can be positive or negative).

Instantaneous velocity

Velocity at a specific moment; equals the slope of the tangent line on an x vs. t graph.

Acceleration (a)

The rate at which velocity changes with time; a vector quantity.

Average acceleration (a_avg)

Change in velocity over change in time: $a_{\text{avg}} = \frac{\Delta v}{\Delta t}$ where $\Delta v = v_f - v_i.$

Instantaneous acceleration

Acceleration at a specific moment; equals the slope of the tangent line on a v vs. t graph.

Uniform (constant) acceleration

Acceleration that stays constant over time, causing velocity to change by equal amounts in equal time intervals.

Non-uniform acceleration

Acceleration that changes over time; often analyzed with graphs or piecewise-constant intervals in AP Physics 1.

Slope (rate of change)

Graph feature that represents how one quantity changes with another (e.g., slope of x–t is v; slope of v–t is a).

Area under a curve (accumulated change)

Graph feature that represents a total change (e.g., area under $v-t$ gives displacement; area under $a-t$ gives $\Delta v$).

Position–time graph (x vs. t)

A graph where slope represents velocity; curved x–t indicates changing velocity (nonzero acceleration).

Velocity–time graph (v vs. t)

A graph where slope represents acceleration and (signed) area under the curve represents displacement.

Acceleration–time graph (a vs. t)

A graph where (signed) area under the curve represents change in velocity: $\Delta v.$

y-intercept (of x vs. t)

The value of position at t = 0, representing the initial position.

Signed area (on v vs. t)

Area under a v–t curve counts direction: negative velocity contributes negative displacement.

Constant-acceleration relation: vf = vi + aΔt

Equation for final velocity when acceleration is constant over the interval.

Constant-acceleration relation: Δx = v_iΔt + (1/2)a(Δt)^2

Equation for displacement when acceleration is constant over the interval.

Constant-acceleration relation: $v_f^2 = v_i^2 + 2a\Delta x$

Time-independent constant-acceleration equation relating velocities, acceleration, and displacement.

Average velocity under constant acceleration

For constant acceleration in 1D, vavg = (vi + v_f)/2.

“Big Five” kinematics equations

A common set of constant-acceleration equations using u (initial v), v (final v), a, s (displacement), and t.

Free fall

Motion under gravity alone (ignoring air resistance), modeled as constant downward acceleration near Earth’s surface.

Acceleration due to gravity (g)

Magnitude of gravitational acceleration near Earth: $g \approx 9.8 \, \text{m/s}^2;$ direction is downward.

Vertical toss peak condition

At the top of a vertical toss, instantaneous velocity v = 0 but acceleration is still downward (a = −g if up is positive).

Horizontal launch (projectile)

A projectile launched with v{y,i} = 0; it immediately gains downward velocity due to gravity while vx stays constant (ideal model).

Independence of perpendicular components

In 2D motion, x- and y-components are independent (except they share the same time variable).

Velocity components (vx, vy)

The horizontal and vertical parts of velocity used to analyze 2D motion separately in each axis.

Component decomposition: v{x,i} = v0 cosθ

Formula for the initial horizontal velocity component of a launch at speed $v_0$ and angle $\theta$.

Component decomposition: v{y,i} = v0 sinθ

Formula for the initial vertical velocity component of a launch at speed $v_0$ and angle $\theta$.

Projectile motion assumptions

Ideal projectile motion near Earth (no air resistance): ax = 0 and ay = −g (if up is positive).

Time of flight (level ground)

For a projectile that lands at its launch height: T = 2v_{y,i}/g (ignoring air resistance).

Range (level ground)

For same launch/landing height: $R = \frac{v_0^2 \sin(2\theta)}{g}$ (ignoring air resistance).

Maximum height (level-ground projectile)

For same launch/landing height: $h = \frac{(v_0 \sin \theta)^2}{2g}$ above the launch point.

Relative velocity (v_{A/B})

Velocity of object A as measured in frame B; depends on the chosen reference frame.

Relative velocity equation (1D)

v{A/B} = v{A/E} − v_{B/E}, where E is a chosen Earth/ground frame.

Frame dependence

Measured kinematics quantities (like velocity) can differ between reference frames, yet both measurements can be correct.

Misconception: negative means slowing down

A negative value usually indicates direction (negative axis), not necessarily that the object is slowing down.

Speeding up condition (v and a)

An object speeds up when velocity and acceleration point in the same direction (same sign in 1D).

Slowing down condition (v and a)

An object slows down when velocity and acceleration point in opposite directions (opposite signs in 1D).

Linearization ($\Delta y$ vs. $t^2$)

For drop-from-rest free fall, plotting displacement $\Delta y$ versus $t^2$ should produce a straight line (constant acceleration model).

Slope meaning in $\Delta y$ vs. $t^2$ free-fall plot

From $\Delta y = \frac{1}{2} g t^2$ (from rest), the slope of $\Delta y$ vs. $t^2$ equals $\frac{1}{2} g$ (sign depends on axis choice).