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Displacement vs. Distance Traveled.
Displacement: Change in position | Distance Traveled: Total path length.
Vector-Valued vs. Parametric Functions.
Vector-Valued: $\mathbf{r}(t) = \langle f(t), g(t) \rangle$ | Parametric: $x(t) = f(t), y(t) = g(t)$
Velocity vs. Speed.
Velocity: Vector with magnitude and direction | Speed: Magnitude of velocity (scalar).
Differentiation vs. Integration.
Differentiation: Finds rate of change | Integration: Finds accumulation/area.
Position vs. Velocity.
Position: Location at time t | Velocity: Rate of change of position at time t
Velocity vs. Acceleration.
Velocity: Rate of change of position at time t | Acceleration: Rate of change of velocity at time t
Average Velocity vs. Instantaneous Velocity.
Average Velocity: Displacement / Time | Instantaneous Velocity: Velocity at a specific time
Average Speed vs. Instantaneous Speed.
Average Speed: Total Distance / Time | Instantaneous Speed: Speed at a specific time
Displacement with Vector Valued Functions vs. Parametric Functions
Vector Valued: Integrate the vector | Parametric: Integrate each component separately
Distance Traveled with Vector Valued Functions vs. Parametric Functions
Vector Valued: Integrate the magnitude of the derivative of the position vector | Parametric: Integrate the square root of the sum of the squares of the derivatives of x and y with respect to t
Define position.
Location of an object at a given time, denoted as $s(t)$.
Define velocity.
Rate of change of position with respect to time; $v(t) = \frac{ds}{dt}$.
Define acceleration.
Rate of change of velocity with respect to time; $a(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2}$.
Define displacement.
Change in position of an object; $\Delta s = s_{final} - s_{initial}$.
Define distance traveled.
Total path length covered by an object.
What is a vector-valued function?
A function represented as $\textbf{r}(t) = \langle f(t), g(t) \rangle$, defining an object's position at time $t$.
What is a parametric function?
Functions where $x$ and $y$ are defined in terms of a parameter $t$: $x(t) = f(t), y(t) = g(t)$.
Define arc length.
The length of a curve.
What does $\mathbf{r}'(t)$ represent?
The derivative of the position vector $\mathbf{r}(t)$, which is the velocity vector.
What does $|\mathbf{r}'(t)|$ represent?
The magnitude of the velocity vector, which is the speed.
How to find displacement given $\mathbf{v}(t)$ from $t=a$ to $t=b$?
1. Integrate $\mathbf{v}(t)$ from $a$ to $b$. 2. Evaluate the integral to find the change in position.
How to find distance traveled given $\mathbf{r}(t)$ from $t=a$ to $t=b$?
1. Find $\mathbf{r}'(t)$. 2. Find $|\mathbf{r}'(t)|$. 3. Integrate $|\mathbf{r}'(t)|$ from $a$ to $b$.
How to find velocity given $\mathbf{r}(t)$?
Differentiate $\mathbf{r}(t)$ with respect to $t$.
How to find acceleration given $\mathbf{v}(t)$?
Differentiate $\mathbf{v}(t)$ with respect to $t$.
How to find distance traveled with parametric equations $x(t)$ and $y(t)$?
1. Find $\frac{dx}{dt}$ and $\frac{dy}{dt}$. 2. Use the formula $S = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt$.
Given position $s(t)$, how do you find the time when the object is at rest?
1. Find $v(t) = s'(t)$. 2. Set $v(t) = 0$ and solve for $t$.
How do you determine if an object is speeding up or slowing down?
1. Find $v(t)$ and $a(t)$. 2. If $v(t)$ and $a(t)$ have the same sign, the object is speeding up. If they have opposite signs, the object is slowing down.
How do you find the average velocity of a particle?
Divide the total displacement by the total time elapsed.
How do you find the average speed of a particle?
Divide the total distance traveled by the total time elapsed.
How do you find the maximum height of a projectile?
1. Find the time when the vertical velocity is zero. 2. Plug that time into the position function to find the height.