Position and Velocity vs Time graphs

A position vs. time graph shows where an object is located at each moment in time. The vertical axis represents position $x$ (measured in meters), and the horizontal axis represents time $t$ (measured in seconds). Every point on the graph tells us exactly where the object is at that specific time.

Straight Lines Mean Constant Velocity

When an object moves with constant velocity, its position graph forms a straight line. The slope of this line reveals two important things:

The magnitude of the velocity: How steep the line is

The direction: Whether the line goes up or down

The velocity is calculated from the slope using:

$$v = \dfrac{\Delta x}{\Delta t} = \dfrac{\text{rise}}{\text{run}} = \dfrac{\text{change in position}}{\text{change in time}}$$

Let's examine three different types of motion:

1. Motion in the Positive Direction

When an object moves forward (positive direction), its position values increase over time. This creates an upward-sloping line on the position graph.

In these graphs, the object starts at the origin ($x_0 = 0\text{ m}$) and moves steadily forward at $v = 4\text{ m/s}$. Notice how:

1. The position line slopes upward (positive slope)

2. Each second, the position increases by $4$ meters

3. After $5$ seconds, the object reaches position $x = 20\text{ m}$

We can calculate velocity:

$$v = \dfrac{\Delta x}{\Delta t} = \dfrac{20 - 0}{5 - 0} = \dfrac{20}{5} = 4\text{ m/s}$$

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The velocity graph shows a horizontal line at $v = 4\text{ m/s}$ (constant positive velocity).

2. Object at Rest

When an object doesn't move, its position stays constant. This creates a horizontal line on the position graph.

Here, the object remains at position $x = 8$ m throughout the entire time period. No change in position means zero velocity.

We can calculate velocity explictly:

$$v = \dfrac{\Delta x}{\Delta t} = \dfrac{8 - 8}{8 - 0} = \dfrac{0}{8} = 0\text{ m/s}$$

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3. Motion in the Negative Direction

When an object moves backward (negative direction), its position values decrease over time. This creates a downward-sloping line on the position graph.