Kinematics: Constant Acceleration and Calculus in One-Dimensional Motion

Vocabulary flashcards covering key concepts from the lecture on constant acceleration, calculus in motion, and historical context.

Acceleration

The rate at which velocity changes with time; measured in meters per second squared (m/s^2); can be positive (speeding up) or negative (slowing down); may be constant or not.

Velocity

The rate of change of position with respect to time; the speed in a given direction; instantaneous velocity is dx/dt, and average velocity is Δx/Δt over an interval.

Instantaneous velocity

The velocity of an object at a specific moment in time.

Delta x

Change in position; Δx = xfinal − xinitial.

Delta t

Change in time; Δt = tfinal − tinitial.

Infinitesimal

An extremely small quantity that is not zero; used in calculus as a limit to zero (e.g., dx and dt).

Derivative

The instantaneous rate of change; for position, the derivative dx/dt gives velocity; for velocity, dv/dt gives acceleration.

Integral

A sum of infinitely many infinitesimals; used to accumulate quantities over an interval (e.g., distance = ∫ v dt).

Constant acceleration

An acceleration that does not change with time; enables simple closed-form expressions for velocity and position.

v(t) = v0 + a t

Velocity at time t for constant acceleration a with initial velocity v0.

x(t) = x0 + v0 t + 1/2 a t^2

Position at time t for constant acceleration a, starting from initial position x0 and initial velocity v0.

v^2 equation

v^2 = v0^2 + 2 a (x − x0); relates velocity to displacement for constant acceleration without explicit time.

Boundary conditions

Known values at specified times or positions (e.g., x(t1) = x1) used to determine constants of integration.

Constant of integration

An added constant from integrating a function; fixed by boundary conditions (e.g., initial position x0 or initial velocity v0).

Initial velocity

The velocity at the starting time (t = 0); often denoted v0, vinitial, or vnaught.

Initial position

The position at the starting time (t = 0); often denoted x0 or x_initial.

Gravity

Earth’s gravitational acceleration; near the surface, approximately 9.8 m/s^2 downward; a standard example of constant acceleration.

g = 9.8 m/s^2

Numerical value of acceleration due to gravity near Earth’s surface.

One-dimensional motion

Motion along a single axis (x); analysis reduces to the x-direction with velocity v and position x.

Commutative property

Order of addition or multiplication does not affect the result (a + b = b + a; a·b = b·a).

Newton

17th-century scientist who developed calculus and applied mathematics to motion; pivotal in the birth of classical physics.

Galileo

Early scientist known for careful experiments and foundational ideas in kinematics; regarded as a precursor to modern science.

Delta notation explanation

Delta indicates a change; Δx = xfinal − xinitial and Δt = tfinal − tinitial.