Vectors and Kinematics Practice Flashcards

Flashcards covering vector notations, operations, calculus derivatives, kinematics equations, and projectile motion based on Unit 1 lecture notes.

Ordered Pair Notation

$$A = (A_x, A_y)$$

Column Vector Notation

$$A = \begin{pmatrix} A_x \\ A_y \end{pmatrix}$$

Unit Vector Notation

$$A = A_x \hat{i} + A_y \hat{j}$$

Polar Notation

$$A = A @ \theta_A$$

Vector Components (Relative to x-axis)

$A_y = |A| \sin(\theta_A)$ and $A_x = |A| \cos(\theta_A)$

Vector Magnitude

$$|A| = \sqrt{A_x^2 + A_y^2 + A_z^2}$$

Vector Direction ($\theta_A$)

$\tan^{-1}\left(\frac{A_y}{A_x}\right)$. Note: If $A_x < 0$, add $180^\circ$ to direction.

Vector Addition and Subtraction Rule

Simply add or subtract like components.

Scalar Multiplication Rule (Magnitude)

If a vector $A$ is multiplied by a scalar $k$, the magnitude $|A|$ is multiplied by the absolute value of the scalar $k$. ($|B| = |kA|$)

Scalar Multiplication Rule (Direction)

If vector $A$ is multiplied by scalar $k$, the direction is unchanged if $k > 0$, or has $180^\circ$ added/subtracted if $k < 0$. ($\theta_B = \theta_A$ for $k > 0$; $\theta_B = \theta_A \pm 180^\circ$ for $k < 0$)

Differentiation

A mathematical operation that acts on a function $y = f(x)$. The result is the "derivative" $f'(x)$, which represents the slope of the function and the instantaneous rate of change at a particular value $x$.

Constant Rule

If $f(x) = c$ (a constant), then $f'(x) = 0$.

Power Rule

If $f(x) = cx^n$ (where $c$ and $n$ are constants and $n \neq 0$), then $f'(x) = cnx^{n-1}$.

Derivative of $\sin(x)$

$$\frac{d}{dx}[\sin(x)] = \cos(x)$$

Derivative of $\cos(x)$

$$\frac{d}{dx}[\cos(x)] = -\sin(x)$$

Derivative of $\ln(x)$

$$\frac{d}{dx}[\ln(x)] = \frac{1}{x}$$

Derivative of $e^x$

$$\frac{d}{dx}[e^x] = e^x$$

Second Derivative

The derivative of the derivative of a function $f(x)$, noted as $f''(x)$ or $\frac{d^2f}{dx^2}$.

Critical Point

The value of $x$ where $f'(x) = 0$. Can represent a local maximum, local minimum, or inflection point.

Local Maximum Condition

Point where the derivative $y' = 0$ and the second derivative $y'' < 0$ (concave down).

Local Minimum Condition

Point where the derivative $y' = 0$ and the second derivative $y'' > 0$ (concave up).

Inflection Point Condition

Point where the function "stops" then "goes" again in the same direction, and where $y' = 0$ and $y'' = 0$.

Instantaneous Velocity ($v$)

$$v = \frac{dx}{dt}$$

Instantaneous Acceleration ($a$)

$$a = \frac{dv}{dt}$$

Average Velocity ($\bar{v}$)

$$\bar{v} = \frac{\Delta x}{\Delta t}$$

Average Acceleration ($\bar{a}$)

$$\bar{a} = \frac{\Delta v}{\Delta t}$$

Constant Acceleration Equation (No time)

$$v^2 = v_0^2 + 2a\Delta x$$

Slope of a Position-Time (x vs. t) Graph

Velocity

Slope of a Velocity-Time (v vs. t) Graph

Acceleration

Area under a Velocity-Time (v vs. t) Graph

Change in position (displacement)

Area under an Acceleration-Time (a vs. t) Graph

Change in velocity

Projectile Motion

Occurs when a particle is in free fall (only gravity acts). Motion in the x-direction is constant velocity, while the y-direction has constant acceleration ($g \approx -10\,m/s^2$).

Time of Flight (T) for Projectile over Level Ground

$$T = \frac{-2v_0 \sin(\theta)}{g}$$

Maximum Height (H) for Projectile over Level Ground

$$H = \frac{v_0^2 \sin^2(\theta)}{2g}$$

Horizontal Range (R) for Projectile over Level Ground

$$R = \frac{-v_0^2 \sin(2\theta)}{g}$$

Reference Frame

A coordinate system that allows an observer to measure and describe the position of an object relative to the observer's own location.

Inertial Reference Frame

A reference frame in which an object either remains at rest or moves at a constant velocity if no net force acts on it.

Non-Inertial Reference Frame

A reference frame that is accelerating, which can lead to the appearance of fictitious forces, such as centrifugal force.

Relative Motion

The calculation of the motion of an object as observed from a particular reference frame.

Velocity of Object in Different Reference Frames

If two reference frames move relative to one another, the velocity of an object in one frame can be found by adding or subtracting the velocity of the frame itself.

Galileo's Principle of Relativity

The laws of physics are the same in all inertial frames of reference.

Transformation of Velocities

The process of converting the velocity of an object in one reference frame to the velocity in another frame by using the relative motion formulas.