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How to find $s(t)$ given $v(t)$ and $s(0)$?
1. 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 $v(0)$?
1. 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 $v(t)$?
1. Evaluate the definite integral $\int_{a}^{b} v(t) dt$.
How to find distance traveled from $t=a$ to $t=b$ given $v(t)$?
1. Evaluate the definite integral $\int_{a}^{b} |v(t)| dt$.
How do you determine when an object changes direction given $v(t)$?
1. 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?
1. Find when $v(t)$ and $a(t)$ have the same sign.
How do you determine when an object is slowing down?
1. 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?
1. 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?
1. 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 $[a,b]$?
1. Calculate the displacement: $\int_{a}^{b} v(t) dt$. 2. Divide the displacement by the time interval $(b-a)$.
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.
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.