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Explain the relationship between position, velocity, and acceleration.
Velocity is the derivative of position, and acceleration is the derivative of velocity. Integration reverses this.
Explain the difference between displacement and distance traveled.
Displacement is the change in position; distance traveled is the total path length.
How is arc length related to distance traveled?
Arc length calculates the total length of the path, giving the distance traveled.
What does integrating a velocity function give you?
Integrating a velocity function gives you the displacement.
What does the magnitude of the velocity vector represent?
The magnitude of the velocity vector represents the speed of the object.
How do you find the velocity vector from a position vector?
Differentiate each component of the position vector with respect to time.
How do you find the acceleration vector from a velocity vector?
Differentiate each component of the velocity vector with respect to time.
What does the definite integral of speed represent?
The total distance traveled over the given time interval.
What is the geometric interpretation of displacement?
The area under the velocity-time curve.
Explain how parametric equations describe motion.
Parametric equations describe motion by defining the x and y coordinates as functions of time, $t$.
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.