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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 $(b-a)$ .

On a position vs. time graph, what does a steeper slope indicate?

A higher velocity.

On a velocity vs. time graph, what does the area above the x-axis represent?

Positive displacement.

On a velocity vs. time graph, what does the area below the x-axis represent?

Negative displacement.

On a velocity vs. time graph, what does a point on the x-axis represent?

The object is momentarily at rest or changing direction.

On an acceleration vs. time graph, what does the area under the curve represent?

The change in velocity.

How do you find the instantaneous velocity at a specific time from a position vs. time graph?

Find the slope of the tangent line at that time.

How do you find the instantaneous acceleration at a specific time from a velocity vs. time graph?

Find the slope of the tangent line at that time.

What does a concave up position vs. time graph signify?

Positive acceleration; the object is speeding up.

What does a concave down position vs. time graph signify?

Negative acceleration; the object is slowing down.

How can you tell from a position vs. time graph if an object is moving towards or away from the origin?

If the absolute value of the position is increasing, it's moving away; if decreasing, it's moving towards.

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