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📙 3.1 Straight Line Motion

The Concepts

In particle motion problems, we have three equations we will work with:

$LaTeX: s(t) \text{ or } x(t) \text{ or } y(t)$ $s(t) \text{ or } x(t) \text{ or } y(t)$ describes the position of the particle, when using $x(t)$ we are looking at the particle moving left and right and for $y(t)$ we are looking at the particle moving up and down.
$v(t) = s'(t)$ describes the velocity of the particle.
$a(t) = v'(t) = s''(t)$ describes the acceleration of the particle.

And we can answer questions about...

Particle Motion Question Types
Statement	Translation	Steps to Solve
The particle has stopped...	when $v(t) = 0$ (i.e. the velocity of the particle is 0).	Set $v(t) = 0$ and solve for $t$ .
The particle is moving to the right (or up)...	when $v(t) > 0$ (i.e. the velocity of the particle is positive).	Set $v(t) > 0$ and solve for $t$ using a sign table or the graph of $v(t)$ .
The particle is moving to the left (or down)...	when $v(t) < 0$ (i.e. the velocity of the particle is negative).	Set $v(t) < 0$ and solve for $t$ using a sign table or the graph of $v(t)$ .
The particle has changed directions or turned around...	when $LaTeX: v(t) \text{ changes signs}$ $v(t) \text{ changes signs}$ (i.e. the velocity of the particle changes from + / -)	Check for sign changes on the sign table or crossing the $t$ -axis for the graph of $v(t)$ .
The particle is speeding up...	when $LaTeX: v(t) \text{ and } a(t) \text{ have the same signs}$ $v(t) \text{ and } a(t) \text{ have the same signs}$ (i.e. when the particle is moving and accelerating in the same direction)	Check signs of $LaTeX: v(t) \text{ and } a(t)$ $v(t) \text{ and } a(t)$ on sign tables of graphs and see where they have the same sign.
The particle is slowing down...	when $LaTeX: v(t) \text{ and } a(t) \text{ have opposite signs}$ $v(t) \text{ and } a(t) \text{ have opposite signs}$ (i.e. when the particle is moving and accelerating in different directions)	Check signs of $LaTeX: v(t) \text{ and } a(t)$ $v(t) \text{ and } a(t)$ on sign tables of graphs and see where they have opposite signs.

Want notes and examples? Here are the completed notes from class: Particle Motion Notes Download Particle Motion Notes. You also have my hand-written, color-coded notes. Links to an external site.

Conceptual Help

Check out this great Geogebra visualization for straight line motion Links to an external site. which I demo in the video below:

Additional Resources

Flipped Math
- 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration Links to an external site.
TBIL Calculus
- 3.1 Tangents motion, and marginals (AD1) Links to an external site.
DeltaMath
- 👩‍🏫 3.1

More Practice

Here are some skill builders: