Topic 1.2 Notes – Displacement, Velocity, and Acceleration

1. The Object Model and Position

Treating objects as point particles

In kinematics, we use the object model. That means:

• Ignore size, shape, and internal motion.
• Treat the object as a single point.
• Keep important properties like mass or charge.

This works when rotation or deformation doesn’t affect what you’re analyzing. A car driving down a straight road can be treated as a point. We only care about its position as a function of time, not how its wheels spin.

Position and coordinate systems

Position tells you where the object is relative to a reference point.

To describe position, you must choose:

• An origin $(x = 0)$
• A positive direction (usually right or up)

Position can be positive, negative, or zero depending on your choice.

If you move the origin, the numbers change, but the physical motion does not. This is something teachers love to test conceptually. The motion is real. The coordinate system is your choice.

2. Displacement

What displacement is

Displacement is the change in position.

$$\Delta x = x_f - x_i$$

• It’s a vector.
• It can be positive, negative, or zero.
• It depends only on initial and final position.

Quick example:
If $x_i = 2\,\text{m}$ and $x_f = -3\,\text{m}$,

$$\Delta x = -3 - 2 = -5\,\text{m}$$

The negative sign tells you the direction.

Displacement vs distance

If you walk 10 m east and 10 m west:

• Displacement = 0
• Distance = 20 m

On MCQs, they love giving motion that reverses direction. Always check whether they want distance or displacement.

Interpreting displacement physically

• The sign tells direction relative to your axis.
• Large displacement does not mean a long path.
• In 1D motion, displacement completely describes how position changed.

That idea leads directly to velocity.

3. Average Velocity

What average velocity means

Average velocity tells you how fast and in what direction position changes over a time interval.

$$v_{\text{avg}} = \frac{\Delta x}{\Delta t}$$

• Vector quantity
• Units: m/s
• Uses total displacement over total time

It only cares about the starting and ending positions.

If $\Delta x = 0$, then $v_{\text{avg}} = 0$, even if the object moved the whole time.

Velocity vs average speed

Students mix this up constantly. Speed ignores direction completely.

Reading velocity from position-time graphs

On a position-time graph, velocity comes from the slope.

Slope on a position-time graph represents velocity

• Average velocity = slope of the secant line between two points on the curve.
• Positive slope → positive velocity.
• Zero slope → object at rest.
• Steeper slope → larger magnitude velocity.

On FRQs, if they ask for “justify,” say: The slope of a position-time graph represents velocity because slope equals change in position over change in time.

4. Average Acceleration

What acceleration is

Acceleration is the rate of change of velocity.

$$a_{\text{avg}} = \frac{\Delta v}{\Delta t}$$

• Vector quantity
• Units: m/s²

An object accelerates if:

• Its speed changes
• Its direction changes
• Or both

Constant velocity means zero acceleration.

Direction of acceleration

Acceleration points in the direction of Δv, not necessarily the direction of motion.

• Velocity becoming more positive → positive acceleration.
• Velocity becoming more negative → negative acceleration.
• Moving right but slowing down → acceleration is left.

“Deceleration” just means acceleration opposite the motion.

Reading acceleration from velocity-time graphs

On a velocity-time graph, acceleration is built into the slope.

Velocity-time graph with varying slopes

• Average acceleration = slope of the graph.
• Horizontal line → zero acceleration.
• Upward slope → positive acceleration.
• Downward slope → negative acceleration.
• Steeper slope → larger acceleration magnitude.

5. Instantaneous vs Average Quantities

Average values use two endpoints.

Instantaneous values describe what’s happening at one moment.

• Instantaneous velocity = slope of tangent on position-time graph.
• Instantaneous acceleration = slope of tangent on velocity-time graph.
• As $\Delta t \to 0$, average values approach instantaneous values.

On the AP exam, they often shrink the time interval and ask what happens to the calculated value. The answer is that it approaches the instantaneous value.