Glossary
Derivative
In calculus, the derivative represents the instantaneous rate of change of a function. For vector-valued functions, the derivative of position yields velocity.
Example:
To find a particle's velocity from its position function, you must take the derivative of each component with respect to time.
Direction of Motion
The direction of motion of a particle is determined by the signs of its horizontal (x'(t)) and vertical (y'(t)) velocity components.
Example:
If x'(t) > 0 and y'(t) < 0, the particle's direction of motion is to the right and downwards.
Horizontal Velocity (x'(t))
The horizontal component of the velocity vector, x'(t), indicates the rate of change of the particle's x-coordinate, determining movement to the left or right.
Example:
If a particle's horizontal velocity is x'(t) = -5, it means the particle is moving 5 units per second to the left.
Magnitude of the Position Vector
The magnitude of the position vector, |p(t)|, represents the distance of the particle from the origin (0,0) at time t.
Example:
If a car's position is p(t) = < 3, 4 >, its magnitude of the position vector is 5, meaning it's 5 units from the origin.
Parametric Functions
Parametric functions define the x and y coordinates of a point in terms of a third independent variable, typically time (t).
Example:
The path of a projectile can be described by parametric functions like x(t) = (V₀cosθ)t and y(t) = (V₀sinθ)t - (1/2)gt².
Planar Motion
Planar motion refers to the movement of an object that is restricted to a two-dimensional plane, such as the xy-plane.
Example:
A robot moving on a flat floor exhibits planar motion as its movement is confined to a single horizontal surface.
Position Vector
The position vector, denoted as p(t), specifies the exact location of a particle at any given time, t, in a 2D plane.
Example:
A drone's position vector might be p(t) = < 5t, -t² + 10t >, showing its location in the sky at time t.
Speed
Speed is the scalar magnitude of the velocity vector, |v(t)|, representing how fast a particle is traveling without regard to its direction.
Example:
If a car's velocity is v(t) = < 3, 4 >, its speed is 5 mph, regardless of its specific direction of travel.
Unit Vectors (i, j)
Unit vectors are vectors of length one that point along the positive x-axis (i) and positive y-axis (j), used to express other vectors in component form.
Example:
The position vector < 3, 4 > can be expressed using unit vectors as *3i + 4j.
Velocity Vector
The velocity vector, denoted as v(t), describes both how fast and in what direction a particle is moving at any time t. It is the derivative of the position vector.
Example:
If a rocket's position is p(t) = < t², t³ >, its velocity vector would be v(t) = < 2t, 3t² >, indicating its instantaneous speed and direction.
Vertical Velocity (y'(t))
The vertical component of the velocity vector, y'(t), indicates the rate of change of the particle's y-coordinate, determining movement upwards or downwards.
Example:
If a particle's vertical velocity is y'(t) = 2, it means the particle is moving 2 units per second upwards.