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What does the area under a velocity-time graph represent?

Displacement.

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What does the area under a velocity-time graph represent?

Displacement.

What does the slope of a position-time graph represent?

Velocity.

What does the slope of a velocity-time graph represent?

Acceleration.

How can you determine when an object changes direction from a velocity-time graph?

Look for points where the velocity graph crosses the t-axis (changes sign).

How can you estimate the total distance traveled from a velocity-time graph?

Estimate the area between the curve and the t-axis, considering areas below the axis as positive.

What does a horizontal line on a velocity-time graph indicate?

Constant velocity (zero acceleration).

What does a horizontal line on an acceleration-time graph indicate?

Constant acceleration.

How does the concavity of a position-time graph relate to acceleration?

Concave up indicates positive acceleration; concave down indicates negative acceleration.

What does the area under an acceleration-time graph represent?

Change in velocity.

How can you determine the time intervals when an object is speeding up or slowing down from a velocity-time graph?

Speeding up when velocity and acceleration have the same sign (both positive or both negative); slowing down when they have opposite signs.

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 $\mathbf{v}(t) = \langle f(t), g(t) \rangle$ ?

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

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

All Flashcards

What does the area under a velocity-time graph represent?

Displacement.

What does the slope of a position-time graph represent?

Velocity.

What does the slope of a velocity-time graph represent?

Acceleration.

How can you determine when an object changes direction from a velocity-time graph?

Look for points where the velocity graph crosses the t-axis (changes sign).

How can you estimate the total distance traveled from a velocity-time graph?

Estimate the area between the curve and the t-axis, considering areas below the axis as positive.

What does a horizontal line on a velocity-time graph indicate?

Constant velocity (zero acceleration).

What does a horizontal line on an acceleration-time graph indicate?

Constant acceleration.

How does the concavity of a position-time graph relate to acceleration?

Concave up indicates positive acceleration; concave down indicates negative acceleration.

What does the area under an acceleration-time graph represent?

Change in velocity.

How can you determine the time intervals when an object is speeding up or slowing down from a velocity-time graph?

Speeding up when velocity and acceleration have the same sign (both positive or both negative); slowing down when they have opposite signs.

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 $\mathbf{v}(t) = \langle f(t), g(t) \rangle$ ?

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

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