Explain the relationship between position and velocity.

Velocity is the derivative of position. It represents the instantaneous rate of change of position.

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Explain the relationship between position and velocity.

Velocity is the derivative of position. It represents the instantaneous rate of change of position.

Explain the relationship between velocity and acceleration.

Acceleration is the derivative of velocity. It represents the instantaneous rate of change of velocity.

What does it mean for an object to be 'speeding up'?

The velocity and acceleration have the same sign (both positive or both negative).

What does it mean for an object to be 'slowing down'?

The velocity and acceleration have opposite signs (one positive and one negative).

Explain the difference between displacement and total distance traveled.

Displacement is the change in position, while total distance traveled considers the absolute value of the velocity, accounting for changes in direction.

How do you determine when a particle changes direction?

The particle changes direction when the velocity changes sign (from positive to negative or vice versa).

What is the significance of finding when $v(t) = 0$ ?

It indicates a potential change in direction or a moment when the object is at rest.

How can the second derivative help in motion problems?

The second derivative, $x''(t)$ , gives the acceleration, which describes how the velocity is changing.

Explain the concept of average velocity.

Average velocity is the change in position divided by the change in time over a given interval.

How does the initial condition relate to finding the position function?

The initial condition, such as $x(0) = c$ , allows you to solve for the constant of integration when finding the position function from the velocity function.

What is the formula for velocity, given position $x(t)$ ?

$v(t) = x'(t) = \frac{dx}{dt}$

What is the formula for acceleration, given velocity $v(t)$ ?

$a(t) = v'(t) = \frac{dv}{dt}$

What is the formula for acceleration, given position $x(t)$ ?

$a(t) = x''(t) = \frac{d^2x}{dt^2}$

How do you find the average velocity over an interval $[a, b]$ ?

$\frac{x(b) - x(a)}{b - a}$

How do you find the instantaneous velocity at time $t=c$ ?

$v(c) = x'(c)$

How do you find the instantaneous acceleration at time $t=c$ ?

$a(c) = v'(c) = x''(c)$

If $v(t)$ is given, how to find the position $x(t)$ ?

$x(t) = \int v(t) dt$

If $a(t)$ is given, how to find the velocity $v(t)$ ?

$v(t) = \int a(t) dt$

How to find the displacement of an object from $t=a$ to $t=b$ ?

$\int_{a}^{b} v(t) dt$

How to find the total distance traveled from $t=a$ to $t=b$ ?

$\int_{a}^{b} |v(t)| dt$

How to find velocity given a position function $x(t)$ ?

Take the derivative of $x(t)$ with respect to $t$ . 2. $v(t) = x'(t)$ .

How to find acceleration given a velocity function $v(t)$ ?

Take the derivative of $v(t)$ with respect to $t$ . 2. $a(t) = v'(t)$ .

How to determine when a particle is moving to the right?

Find the velocity function $v(t)$ . 2. Solve for when $v(t) > 0$ .

How to determine when a particle is moving to the left?

Find the velocity function $v(t)$ . 2. Solve for when $v(t) < 0$ .

How to determine when a particle is speeding up?

Find $v(t)$ and $a(t)$ . 2. Determine when $v(t)$ and $a(t)$ have the same sign.

How to determine when a particle is slowing down?

Find $v(t)$ and $a(t)$ . 2. Determine when $v(t)$ and $a(t)$ have opposite signs.

How to find the displacement of a particle over an interval $[a, b]$ ?

Find the velocity function $v(t)$ . 2. Evaluate $\int_{a}^{b} v(t) dt$ .

How to find the total distance traveled by a particle over an interval $[a, b]$ ?

Find the velocity function $v(t)$ . 2. Evaluate $\int_{a}^{b} |v(t)| dt$ .

How to find the position function $x(t)$ given $v(t)$ and $x(0)$ ?

Integrate $v(t)$ to find $x(t) = \int v(t) dt + C$ . 2. Use the initial condition $x(0)$ to solve for $C$ .

How to find the acceleration at a specific time $t=c$ given $x(t)$ ?

Find the second derivative $a(t) = x''(t)$ . 2. Evaluate $a(c)$ .

All Flashcards

Explain the relationship between position and velocity.

Velocity is the derivative of position. It represents the instantaneous rate of change of position.

Explain the relationship between velocity and acceleration.

Acceleration is the derivative of velocity. It represents the instantaneous rate of change of velocity.

What does it mean for an object to be 'speeding up'?

The velocity and acceleration have the same sign (both positive or both negative).

What does it mean for an object to be 'slowing down'?

The velocity and acceleration have opposite signs (one positive and one negative).

Explain the difference between displacement and total distance traveled.

Displacement is the change in position, while total distance traveled considers the absolute value of the velocity, accounting for changes in direction.

How do you determine when a particle changes direction?

The particle changes direction when the velocity changes sign (from positive to negative or vice versa).

What is the significance of finding when $v(t) = 0$ ?

It indicates a potential change in direction or a moment when the object is at rest.

How can the second derivative help in motion problems?

The second derivative, $x''(t)$ , gives the acceleration, which describes how the velocity is changing.

Explain the concept of average velocity.

Average velocity is the change in position divided by the change in time over a given interval.

How does the initial condition relate to finding the position function?

The initial condition, such as $x(0) = c$ , allows you to solve for the constant of integration when finding the position function from the velocity function.

What is the formula for velocity, given position $x(t)$ ?

$v(t) = x'(t) = \frac{dx}{dt}$

What is the formula for acceleration, given velocity $v(t)$ ?

$a(t) = v'(t) = \frac{dv}{dt}$

What is the formula for acceleration, given position $x(t)$ ?

$a(t) = x''(t) = \frac{d^2x}{dt^2}$

How do you find the average velocity over an interval $[a, b]$ ?

$\frac{x(b) - x(a)}{b - a}$

How do you find the instantaneous velocity at time $t=c$ ?

$v(c) = x'(c)$

How do you find the instantaneous acceleration at time $t=c$ ?

$a(c) = v'(c) = x''(c)$

If $v(t)$ is given, how to find the position $x(t)$ ?

$x(t) = \int v(t) dt$

If $a(t)$ is given, how to find the velocity $v(t)$ ?

$v(t) = \int a(t) dt$

How to find the displacement of an object from $t=a$ to $t=b$ ?

$\int_{a}^{b} v(t) dt$

How to find the total distance traveled from $t=a$ to $t=b$ ?

$\int_{a}^{b} |v(t)| dt$

How to find velocity given a position function $x(t)$ ?

Take the derivative of $x(t)$ with respect to $t$ . 2. $v(t) = x'(t)$ .

How to find acceleration given a velocity function $v(t)$ ?

Take the derivative of $v(t)$ with respect to $t$ . 2. $a(t) = v'(t)$ .

How to determine when a particle is moving to the right?

Find the velocity function $v(t)$ . 2. Solve for when $v(t) > 0$ .

How to determine when a particle is moving to the left?

Find the velocity function $v(t)$ . 2. Solve for when $v(t) < 0$ .

How to determine when a particle is speeding up?

Find $v(t)$ and $a(t)$ . 2. Determine when $v(t)$ and $a(t)$ have the same sign.

How to determine when a particle is slowing down?

Find $v(t)$ and $a(t)$ . 2. Determine when $v(t)$ and $a(t)$ have opposite signs.

How to find the displacement of a particle over an interval $[a, b]$ ?

Find the velocity function $v(t)$ . 2. Evaluate $\int_{a}^{b} v(t) dt$ .

How to find the total distance traveled by a particle over an interval $[a, b]$ ?

Find the velocity function $v(t)$ . 2. Evaluate $\int_{a}^{b} |v(t)| dt$ .

How to find the position function $x(t)$ given $v(t)$ and $x(0)$ ?

Integrate $v(t)$ to find $x(t) = \int v(t) dt + C$ . 2. Use the initial condition $x(0)$ to solve for $C$ .

How to find the acceleration at a specific time $t=c$ given $x(t)$ ?

Find the second derivative $a(t) = x''(t)$ . 2. Evaluate $a(c)$ .