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Formula for velocity given position.

$v(t) = \frac{d}{dt}s(t) = s'(t)$

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Formula for velocity given position.

$v(t) = \frac{d}{dt}s(t) = s'(t)$

Formula for acceleration given velocity.

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

Formula for acceleration given position.

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

Formula for position given velocity.

$s(t) = \int v(t) dt + C$

Formula for velocity given acceleration.

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

Formula for displacement.

$\Delta s = s_f - s_i = \int_{t_i}^{t_f} v(t) dt$

Formula for distance traveled.

$\int_{t_i}^{t_f} |v(t)| dt$

How to find final position?

$s(t_f) = s(t_i) + \int_{t_i}^{t_f} v(t) dt$

How to find final velocity?

$v(t_f) = v(t_i) + \int_{t_i}^{t_f} a(t) dt$

What is the relationship between displacement and velocity?

Displacement is the integral of velocity over a time interval.

How to find $s(t)$ given $v(t)$ and $s(0)$ ?

Integrate $v(t)$ to find the general form of $s(t)$ . 2. Use $s(0)$ to solve for the constant of integration. 3. Write the specific equation for $s(t)$ .

How to find $v(t)$ given $a(t)$ and $v(0)$ ?

Integrate $a(t)$ to find the general form of $v(t)$ . 2. Use $v(0)$ to solve for the constant of integration. 3. Write the specific equation for $v(t)$ .

How to find displacement from $t=a$ to $t=b$ given $v(t)$ ?

Evaluate the definite integral $\int_{a}^{b} v(t) dt$ .

How to find distance traveled from $t=a$ to $t=b$ given $v(t)$ ?

Evaluate the definite integral $\int_{a}^{b} |v(t)| dt$ .

How do you determine when an object changes direction given $v(t)$ ?

Find when $v(t) = 0$ . 2. Check if the sign of $v(t)$ changes around those points.

How do you determine when an object is speeding up?

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

How do you determine when an object is slowing down?

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

Given $v(t)$ and an interval $[a, b]$ , how do you find the maximum position?

Find critical points by setting $v(t) = 0$ . 2. Evaluate $s(t)$ at critical points and endpoints. 3. Choose the largest value.

How do you solve for the total distance traveled when $v(t)$ changes sign on the interval?

Find the times when $v(t) = 0$ . 2. Break the integral into subintervals based on these times. 3. Integrate $|v(t)|$ over each subinterval and add the results.

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

Calculate the displacement: $\int_{a}^{b} v(t) dt$ . 2. Divide the displacement by the time interval $(b-a)$ .

What are the differences between displacement and distance traveled?

Displacement: Change in position, can be negative | Distance Traveled: Total path length, always non-negative.

What are the differences between velocity and speed?

Velocity: Rate of change of position with direction (can be negative) | Speed: Magnitude of velocity (always non-negative).

Compare finding displacement using definite integrals vs. indefinite integrals.

Definite Integral: Directly calculates displacement over an interval | Indefinite Integral: Gives a general position function, requires initial position to find displacement.

Compare finding velocity using derivatives vs. integrals.

Derivatives: Find velocity from position function | Integrals: Find velocity from acceleration function.

Compare average velocity and instantaneous velocity.

Average Velocity: Displacement over a time interval | Instantaneous Velocity: Velocity at a specific moment in time.

Compare positive and negative acceleration.

Positive Acceleration: Velocity is increasing | Negative Acceleration: Velocity is decreasing (deceleration).

Compare constant velocity and constant acceleration.

Constant Velocity: Acceleration is zero, object moves at a steady rate | Constant Acceleration: Velocity changes at a steady rate.

Compare the effect of integrating velocity when $v(t)>0$ vs $v(t)<0$ .

v(t) > 0: Displacement is positive, object moves in positive direction | v(t) < 0: Displacement is negative, object moves in negative direction.

Compare the meaning of a zero velocity and a zero acceleration.

Zero Velocity: Object is momentarily at rest | Zero Acceleration: Velocity is constant.

Compare the use of initial position and initial velocity in solving motion problems.

Initial Position: Used to find the constant of integration when integrating velocity | Initial Velocity: Used to find the constant of integration when integrating acceleration.

All Flashcards

Formula for velocity given position.

$v(t) = \frac{d}{dt}s(t) = s'(t)$

Formula for acceleration given velocity.

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

Formula for acceleration given position.

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

Formula for position given velocity.

$s(t) = \int v(t) dt + C$

Formula for velocity given acceleration.

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

Formula for displacement.

$\Delta s = s_f - s_i = \int_{t_i}^{t_f} v(t) dt$

Formula for distance traveled.

$\int_{t_i}^{t_f} |v(t)| dt$

How to find final position?

$s(t_f) = s(t_i) + \int_{t_i}^{t_f} v(t) dt$

How to find final velocity?

$v(t_f) = v(t_i) + \int_{t_i}^{t_f} a(t) dt$

What is the relationship between displacement and velocity?

Displacement is the integral of velocity over a time interval.

How to find $s(t)$ given $v(t)$ and $s(0)$ ?

Integrate $v(t)$ to find the general form of $s(t)$ . 2. Use $s(0)$ to solve for the constant of integration. 3. Write the specific equation for $s(t)$ .

How to find $v(t)$ given $a(t)$ and $v(0)$ ?

Integrate $a(t)$ to find the general form of $v(t)$ . 2. Use $v(0)$ to solve for the constant of integration. 3. Write the specific equation for $v(t)$ .

How to find displacement from $t=a$ to $t=b$ given $v(t)$ ?

Evaluate the definite integral $\int_{a}^{b} v(t) dt$ .

How to find distance traveled from $t=a$ to $t=b$ given $v(t)$ ?

Evaluate the definite integral $\int_{a}^{b} |v(t)| dt$ .

How do you determine when an object changes direction given $v(t)$ ?

Find when $v(t) = 0$ . 2. Check if the sign of $v(t)$ changes around those points.

How do you determine when an object is speeding up?

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

How do you determine when an object is slowing down?

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

Given $v(t)$ and an interval $[a, b]$ , how do you find the maximum position?

Find critical points by setting $v(t) = 0$ . 2. Evaluate $s(t)$ at critical points and endpoints. 3. Choose the largest value.

How do you solve for the total distance traveled when $v(t)$ changes sign on the interval?

Find the times when $v(t) = 0$ . 2. Break the integral into subintervals based on these times. 3. Integrate $|v(t)|$ over each subinterval and add the results.

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

Calculate the displacement: $\int_{a}^{b} v(t) dt$ . 2. Divide the displacement by the time interval $(b-a)$ .

What are the differences between displacement and distance traveled?

Displacement: Change in position, can be negative | Distance Traveled: Total path length, always non-negative.

What are the differences between velocity and speed?

Velocity: Rate of change of position with direction (can be negative) | Speed: Magnitude of velocity (always non-negative).

Compare finding displacement using definite integrals vs. indefinite integrals.

Definite Integral: Directly calculates displacement over an interval | Indefinite Integral: Gives a general position function, requires initial position to find displacement.

Compare finding velocity using derivatives vs. integrals.

Derivatives: Find velocity from position function | Integrals: Find velocity from acceleration function.

Compare average velocity and instantaneous velocity.

Average Velocity: Displacement over a time interval | Instantaneous Velocity: Velocity at a specific moment in time.

Compare positive and negative acceleration.

Positive Acceleration: Velocity is increasing | Negative Acceleration: Velocity is decreasing (deceleration).

Compare constant velocity and constant acceleration.

Constant Velocity: Acceleration is zero, object moves at a steady rate | Constant Acceleration: Velocity changes at a steady rate.

Compare the effect of integrating velocity when $v(t)>0$ vs $v(t)<0$ .

v(t) > 0: Displacement is positive, object moves in positive direction | v(t) < 0: Displacement is negative, object moves in negative direction.

Compare the meaning of a zero velocity and a zero acceleration.

Zero Velocity: Object is momentarily at rest | Zero Acceleration: Velocity is constant.

Compare the use of initial position and initial velocity in solving motion problems.

Initial Position: Used to find the constant of integration when integrating velocity | Initial Velocity: Used to find the constant of integration when integrating acceleration.