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Explain how integration is used to find position from velocity.

Integrating $v(t)$ gives the change in position (displacement). Add the initial position to find the final position.

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Explain how integration is used to find position from velocity.

Integrating $v(t)$ gives the change in position (displacement). Add the initial position to find the final position.

Explain how integration is used to find velocity from acceleration.

Integrating $a(t)$ gives the change in velocity. Add the initial velocity to find the final velocity.

What does the area under the velocity vs. time curve represent?

The area under the $v(t)$ curve represents the displacement of the object.

What does the area under the absolute value of velocity vs. time curve represent?

The area under the $|v(t)|$ curve represents the total distance traveled by the object.

Why is absolute value important when finding distance traveled?

It accounts for changes in direction, ensuring all movement contributes positively to the total distance.

Explain the significance of the constant of integration, C, when integrating velocity or acceleration.

C represents the initial condition (position or velocity) needed to find the specific function.

Describe the relationship between a position vs. time graph and a velocity vs. time graph.

The slope of the position vs. time graph at any point gives the velocity at that time.

Describe the relationship between a velocity vs. time graph and an acceleration vs. time graph.

The slope of the velocity vs. time graph at any point gives the acceleration at that time.

What does a horizontal line on a position vs. time graph indicate?

The object is at rest; its velocity is zero.

What does a horizontal line on a velocity vs. time graph indicate?

The object is moving at a constant velocity; its acceleration is zero.

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.

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 $(b-a)$ .