Motion in Two Dimensions and Circular Motion - Kinematics (Vocabulary Cards)

Vocabulary flashcards covering key concepts in two-dimensional motion, projectile motion, and circular motion from the notes.

Position Vector (r)

A vector from the origin to the particle’s location in the plane; in components r = x i + y j.

Displacement (Δr)

The change in position; the difference between final and initial position vectors: Δr = rf − ri.

Trajectory

The path that a particle follows in the plane (xy-plane).

Average Velocity (v_avg)

Displacement divided by the time interval: v_avg = Δr/Δt; points in the direction of the displacement.

Instantaneous Velocity

The limit of average velocity as Δt → 0; the velocity at a specific instant; tangent to the trajectory.

Speed

Magnitude of velocity; the rate of motion without regard to direction.

Velocity Components (vx, vy)

Projections of velocity along the x and y axes; vx = dx/dt, vy = dy/dt.

Average Acceleration (a_avg)

Change in velocity over a time interval: a_avg = Δv/Δt.

Instantaneous Acceleration

The limit of average acceleration as Δt → 0; the acceleration at a specific time.

Acceleration Components (ax, ay)

Decomposition of acceleration along the x and y axes; ax = dvx/dt, ay = dvy/dt.

Projectile Motion

Two-dimensional motion under gravity with negligible air resistance; x-acceleration = 0, y-acceleration = −g.

Launch Angle (θ)

Angle of the initial velocity relative to the +x axis.

Initial Velocity (v0)

Velocity at launch; magnitude v0 with components v0x = v0 cosθ and v0y = v0 sinθ.

Independence of Motions

Motion along x and y are independent; x-equations involve x quantities and y-equations involve y quantities.

Uniform Circular Motion (UCM)

Motion around a circle at constant speed; velocity direction changes, producing centripetal acceleration toward the center.

Centripetal Acceleration (ac or ar)

Acceleration toward the circle’s center; magnitude a_r = v^2/r = ω^2 r.

Radius (R)

Distance from the center of the circle to a point on the circular path.

Period (T)

Time for one complete revolution; T = 2πR/v = 2π/ω.

Frequency (f)

Revolutions per unit time; f = 1/T; related to angular velocity by ω = 2π f.

Angular Position (θ)

Angle from the +x axis to the position on the circle; measured in radians, positive for counterclockwise.

Arc Length (s)

Distance traveled along the circular path; s = R θ.

Angular Displacement (Δθ)

Change in angular position; Δθ = θf − θi.

Angular Velocity (ω)

Rate of change of angular position; ω = dθ/dt; units rad/s; constant in uniform circular motion.

Tangential Acceleration (a_t)

Component of acceleration parallel to velocity; changes speed; a_t = α R where α = dω/dt.

Angular Acceleration (α)

Rate of change of angular velocity; α = dω/dt; units rad/s^2; describes speeding up or slowing down of rotation.

Radian (rad)

Unit of angular measure; 1 radian is the angle subtended by an arc length equal to the radius.

Relation v = ω r

Linear speed equals angular speed times radius: v = ω r.

Parabolic Trajectory

The projectile’s path is a parabola obtained by eliminating time between x(t) and y(t) equations.

Gravity (g)

Acceleration due to gravity; downward acceleration g ≈ 9.8 m/s^2 (downward).

Horizontal Motion in Projectile

No horizontal acceleration; x = v0x t.

Vertical Motion in Projectile

Constant downward acceleration −g; y = v0y t − 1/2 g t^2.

Independent Motion Example

Horizontal motion is independent of vertical motion, as seen when a ball is projected horizontally.

Trajectory Equation (y(x))

The relation between y and x for a projectile, typically a parabola described by y(x) = x tanθ − (g x^2)/(2 v0^2 cos^2 θ).

Center-Seeking Acceleration

Another name for centripetal (radial) acceleration toward the circle’s center.

Tangential vs Radial Components

Tangential acceleration changes speed (along v); radial (centripetal) acceleration changes direction (toward center).

Constant Acceleration Kinematics (2D)

When acceleration is constant, x and y motions can be treated independently with Δt the same for both.