Formula for velocity given position $s(t)$ .

$v(t) = \frac{ds}{dt}$

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Formula for velocity given position $s(t)$ .

$v(t) = \frac{ds}{dt}$

Formula for acceleration given velocity $v(t)$ .

$a(t) = \frac{dv}{dt}$

Formula for displacement.

$\Delta s = s_{final} - s_{initial}$

Arc length formula for vector-valued functions.

$S = \int_a^b |\mathbf{r}'(t)| dt$

Arc length formula for parametric functions.

$S = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt$

How do you find displacement given a velocity vector $\mathbf{v}(t)$ ?

$\Delta \mathbf{r} = \int_{t_1}^{t_2} \mathbf{v}(t) dt$

How to find the magnitude of a vector $\langle x, y \rangle$ ?

$\sqrt{x^2 + y^2}$

Formula for acceleration given position $s(t)$ .

$a(t) = \frac{d^2s}{dt^2}$

How do you find the velocity vector $\mathbf{v}(t)$ given the position vector $\mathbf{r}(t) = \langle f(t), g(t) \rangle$ ?

$\mathbf{v}(t) = \langle f'(t), g'(t) \rangle$

How do you find the acceleration vector $\mathbf{a}(t)$ given the velocity vector $\mathbf{v}(t) = \langle f(t), g(t) \rangle$ ?

$\mathbf{a}(t) = \langle f'(t), g'(t) \rangle$

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, $t$ .

How to find displacement given $\mathbf{v}(t)$ from $t=a$ to $t=b$ ?

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 $t=b$ ?

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 $y(t)$ ?

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?

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?

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?

Find the time when the vertical velocity is zero. 2. Plug that time into the position function to find the height.

All Flashcards

Formula for velocity given position $s(t)$ .

$v(t) = \frac{ds}{dt}$

Formula for acceleration given velocity $v(t)$ .

$a(t) = \frac{dv}{dt}$

Formula for displacement.

$\Delta s = s_{final} - s_{initial}$

Arc length formula for vector-valued functions.

$S = \int_a^b |\mathbf{r}'(t)| dt$

Arc length formula for parametric functions.

$S = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} dt$

How do you find displacement given a velocity vector $\mathbf{v}(t)$ ?

$\Delta \mathbf{r} = \int_{t_1}^{t_2} \mathbf{v}(t) dt$

How to find the magnitude of a vector $\langle x, y \rangle$ ?

$\sqrt{x^2 + y^2}$

Formula for acceleration given position $s(t)$ .

$a(t) = \frac{d^2s}{dt^2}$

How do you find the velocity vector $\mathbf{v}(t)$ given the position vector $\mathbf{r}(t) = \langle f(t), g(t) \rangle$ ?

$\mathbf{v}(t) = \langle f'(t), g'(t) \rangle$

How do you find the acceleration vector $\mathbf{a}(t)$ given the velocity vector $\mathbf{v}(t) = \langle f(t), g(t) \rangle$ ?

$\mathbf{a}(t) = \langle f'(t), g'(t) \rangle$

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, $t$ .

How to find displacement given $\mathbf{v}(t)$ from $t=a$ to $t=b$ ?

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 $t=b$ ?

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 $y(t)$ ?

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?

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?

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?

Find the time when the vertical velocity is zero. 2. Plug that time into the position function to find the height.