AP Calc motion along a line cheat sheet

Average velocity formula?

Average velocity $\displaystyle v_{avg} = \\frac{\Delta s}{\Delta t}$ where $\displaystyle \Delta s$ is displacement and $\displaystyle \Delta t$ is time.

Find the velocity at time t.

Given position function $\displaystyle s(t) = 3t^2 + 2t$, then $\displaystyle v(t) = s'(t) = 6t + 2$. Substitute $\displaystyle t = 2$: $\displaystyle v(2) = 14$ ft/sec.

True or False: Acceleration is the derivative of position.

False. Acceleration is the derivative of velocity.

Difference between speed and velocity?

Speed is scalar (magnitude only). Velocity is vector (magnitude and direction). Example: 50 mph East.

If s(t) = t^3 - 6t^2 + 9t, find acceleration.

First, find velocity $\displaystyle v(t) = s'(t) = 3t^2 - 12t + 9$. Then, $\displaystyle a(t) = v'(t) = 6t - 12$.

What happens when acceleration is zero?

An object is at rest or moving at constant velocity (no change in speed/direction).

Calculate displacement for $\displaystyle t=1$ to $\displaystyle t=3$.

For $\displaystyle s(t) = 2t^2 + 3$, $\displaystyle \Delta s = s(3) - s(1) = (2(3)^2 + 3) - (2(1)^2 + 3) = 15 - 5 = 10$.

Fill in the blank: The ______ of an object is its change in position over a time interval.

displacement

What is the relationship between position, velocity, and acceleration?

Position is the integral of velocity. Velocity is the integral of acceleration. Example: $\displaystyle s(t) = \int v(t) \, dt$.

True or False: Velocity can be negative.

True. Negative velocity indicates motion in the opposite direction.

Example of uniform motion.

Driving 60 miles/hour on a straight highway for 2 hours. Total distance: $\displaystyle 120$ miles.

Acceleration when speed increases?

Positive acceleration indicates increasing speed. Example: A car speeds up from 30 mph to 50 mph.

Position function example.

If a car travels $\displaystyle s(t) = 5t^2 + 2t$ miles, find the position at $\displaystyle t=3$ seconds. $\displaystyle s(3) = 5(3^2) + 2(3) = 57$ miles.

Velocity vs. Speed.

Velocity is direction-aware; speed is magnitude only. E.g., traveling 60 mph north (velocity) vs. 60 mph (speed).

Acceleration formula.

Acceleration is the derivative of velocity: $\displaystyle a(t) = v'(t)$. If $\displaystyle v(t) = 4t + 3$, then $\displaystyle a(t) = 4$.

True or False: Constant velocity implies zero acceleration.

True. If velocity does not change, acceleration is zero.

Fill in the blank: The derivative of position is _____ .

velocity. $\displaystyle v(t) = s'(t)$

Find the acceleration at $\displaystyle t=2$ for $\displaystyle s(t)=3t^3-6t^2$.

First find $\displaystyle v(t) = s'(t) = 9t^2 - 12t$; then $\displaystyle a(t) = v'(t) = 18t - 12$. At $\displaystyle t=2$: $\displaystyle a(2) = 24$.

What does the second derivative represent?

The second derivative indicates acceleration. For position $\displaystyle s(t)$, $\displaystyle a(t) = s''(t)$.

Distance traveled vs. Displacement.

Distance is total path length; displacement is the final position minus the initial position. E.g., from 0 to 5 miles and back to 2 miles gives a distance of 7 miles, displacement of 2 miles.

Find velocity from $\displaystyle s(t) = 2t^2 + 4t$.

The velocity is $\displaystyle v(t) = s'(t) = 4t + 4$. At $\displaystyle t=1$, $\displaystyle v(1) = 8$.

Cause → Effect: Increasing speed.

Increasing speed leads to increased distance over time. E.g., doubling speed from 30 mph to 60 mph reduces travel time for a fixed distance.

Examples of increasing/decreasing intervals.

If $\displaystyle v(t)$ is positive, the object moves forward (increasing); if negative, it moves backward (decreasing).

True or False: A zero velocity indicates rest.

False. Zero velocity means no change in position, but object could be changing direction (e.g. turning around).