Topic 1.3 Notes – Representing Motion

1. Position, Velocity, and Acceleration

Position $x$

Position tells you where an object is relative to an origin.

• Includes direction. The sign matters.
• Displacement is change in position: $$\Delta x = x - x_0$$
• Displacement is not total distance traveled.

If you walk 5 m right and 5 m left, your distance is 10 m but your displacement is 0 m.

Velocity $v$

Velocity describes how position changes.

• Average velocity $$v_{\text{avg}} = \frac{\Delta x}{\Delta t}$$
• Instantaneous velocity is the slope of a tangent line on an $x$-$t$ graph.
• The sign tells direction.

Positive velocity means motion in the positive direction. Negative velocity means motion in the negative direction.

Acceleration $a$

Acceleration describes how velocity changes.

• Average acceleration $$a_{\text{avg}} = \frac{\Delta v}{\Delta t}$$
• Instantaneous acceleration is the slope of a tangent line on a $v$-$t$ graph.

Slowing down depends on direction:

• If $v$ and $a$ have the same sign, speed increases.
• If they have opposite signs, speed decreases.

That idea shows up constantly in conceptual questions.

2. Ways to Represent Motion

You’re expected to translate between all of these.

Motion Diagrams

Dots represent position at equal time intervals.

• Equal spacing → constant velocity
• Increasing spacing → speeding up
• Decreasing spacing → slowing down
• Arrow direction → direction of velocity

If arrows get longer, speed is increasing.

Graphs

Understanding slopes and areas is huge.

Position-Time Graph $x$-$t$

• Slope = velocity
• Horizontal line → at rest
• Curved graph → acceleration present

Above the axis gives positive displacement. Below gives negative displacement.

Acceleration-Time Graph $a$-$t$

• Area under curve = change in velocity $\Delta v$
• Horizontal line → constant acceleration
• Zero line → constant velocity

AP Physics 1 does not require calculating with changing acceleration, but you must interpret curved graphs qualitatively.

3. Constant Acceleration Kinematic Equations

These work only in one dimension and only when acceleration is constant.

$$v = v_0 + at$$

$$x = x_0 + v_0 t + \tfrac{1}{2} a t^2$$

$$v^2 = v_0^2 + 2a(x - x_0)$$

Use whichever equation contains only one unknown.

Example idea: If something stops, set $v = 0$. If it’s dropped, $v_0 = 0$. Always define a coordinate system first. Most sign mistakes come from skipping that step.

These equations apply vertically too. Just switch to $y$.

4. Acceleration Due to Gravity

Near Earth:

$$g = 10 \text{ m/s}^2 \text{ downward}$$

• Same for all objects (ignoring air resistance).
• Constant the entire motion.
• Acts only vertically.

If upward is positive, $a = -10$. If downward is positive, $a = +10$.

At maximum height:

• $v = 0$
• $a \neq 0$

If an object lands at the same height it was launched:

• Time up = time down
• Final speed = initial speed (opposite direction)

Students often think acceleration becomes zero at the top. It does not.

5. Graph Relationships Summary

Slope always means “rate of change.”
Area always means “accumulated change.”

6. Nonuniform Acceleration (Qualitative)

You won’t calculate with variable acceleration, but you must interpret it.

• Curved $x$-$t$ graph → changing velocity
• Curved $v$-$t$ graph → changing acceleration
• Increasing slope magnitude → speeding up
• Decreasing slope magnitude → slowing down

On FRQs, you might be asked to sketch a reasonable graph. Make sure slopes and areas match your description of motion.