Representing and Predicting Motion

Driving question

How can we describe and predict motion without explaining what causes it?

Start with a story, not an equation

A cart rolls along a straight track. It passes the origin, continues in the positive direction, slows, stops, and reverses. Before calculating anything, sketch five dots showing where the cart might be at equal time intervals. Add velocity arrows.

This is kinematics: describing where an object is, how its position changes, and how rapidly its velocity changes. A strong model connects four representations:

1. words;
2. a motion diagram;
3. graphs;
4. equations.

Position, displacement, and distance

Position, written as $x$, locates an object relative to an origin. Displacement is a change in position:

Distance is the total path length. It is never negative. Displacement can be positive, negative, or zero because it includes direction.

Example: A student walks from $x=2\text{ m}$ to $x=9\text{ m}$, then back to $x=5\text{ m}$. The distance is $11\text{ m}$, but the displacement is $+3\text{ m}$.

Velocity is the slope of position

Average velocity is

On a position-versus-time graph, average velocity is the slope of a secant line. Instantaneous velocity is the slope of the tangent line at one moment.

• Positive slope means positive velocity.
• Negative slope means negative velocity.
• Zero slope means the object is momentarily at rest.
• A steeper slope means greater speed.

A position graph is not a drawing of the path. Its vertical coordinate reports position; its slope reports velocity.

Acceleration is the slope of velocity

Average acceleration is

On a velocity-versus-time graph, acceleration is the slope. The signed area between the velocity curve and the time axis is displacement.

An object speeds up when velocity and acceleration have the same sign. It slows down when their signs differ. Negative acceleration does not automatically mean slowing down.

Constant-acceleration model

When acceleration is constant:

Choose an equation only after defining the system, axis, origin, initial moment, and known quantities. These equations are not valid when acceleration changes substantially.

Worked example: braking bicycle

A bicycle moving at $+8.0\text{ m/s}$ accelerates uniformly at $-2.0\text{ m/s}^2$. How long does it take to stop, and how far does it travel?

Using $v=v_0+at$:

Then

Checks: seconds are the correct time unit; displacement is positive because the bicycle continues forward; the average velocity is $4.0\text{ m/s}$, so $(4.0)(4.0)=16\text{ m}$.

Two-dimensional motion

Treat perpendicular components independently while using the same clock. For ideal projectile motion with negligible air resistance:

• horizontal acceleration is zero;
• vertical acceleration is approximately $-g$ when positive is upward;
• horizontal and vertical velocities combine as vectors.

At the highest point, vertical velocity is zero for an instant, but acceleration is still downward.

Evidence routine for any motion problem

1. Define the object or system.
2. Choose axes and state the positive direction.
3. Draw a motion diagram.
4. Sketch the relevant graphs.
5. Identify the interval and model assumptions.
6. Write relationships symbolically before substituting values.
7. Check units, signs, graph shape, and limiting behavior.

Misconception clinic

“Velocity and acceleration always point together.” They point together while speeding up and oppositely while slowing down.

“Negative means slowing down.” A negative sign indicates direction relative to the chosen axis.

“A motion graph shows the path.” A graph shows how one measured quantity depends on another.

Retrieval check

1. What does the slope of a position-time graph represent?
2. What does the signed area under a velocity-time graph represent?
3. Can an object have zero velocity and nonzero acceleration? Give an example.
4. Which assumptions make the constant-acceleration equations valid?

Key takeaway

Representations are evidence. When the story, diagram, graph, and equation agree—including their signs and units—the prediction is defensible.