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.

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

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

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.

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

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.