2.3 Position vs. Time Graphs

Position vs. Time Graphs

Position-time graphs show where an object is relative to a reference point as time passes. Position goes on the y-axis and time on the x-axis. The shape of the graph tells you everything about how the object moves, and the slope tells you how fast.

These graphs are your primary tool for finding both average and instantaneous velocity, which connect directly to the slope of the graph line.

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Interpretation of Position-Time Graphs

Every position-time graph uses the same setup: position (in meters, feet, etc.) on the vertical axis and time (in seconds, minutes, etc.) on the horizontal axis. What matters most is the slope and shape of the line.

Straight lines mean constant velocity:

• Positive slope: the object moves at constant velocity in the positive direction (away from the reference point)
• Negative slope: the object moves at constant velocity in the negative direction (toward the reference point)
• Zero slope (horizontal line): the object is at rest and its position isn't changing

Curved lines mean the velocity is changing, so the object is accelerating:

• Concave up (slope getting steeper): the object is speeding up. Think of a car merging onto a highway. The position changes more and more with each passing second.
• Concave down (slope getting less steep): the object is slowing down. Think of a car braking to a stop. The position changes less and less over time.

A common mistake is confusing "position" with "distance traveled." If the graph dips below the x-axis, that just means the object is on the negative side of the reference point. It doesn't mean the object has gone a negative distance.

Average Velocity from Position-Time Graphs

Average velocity is the overall rate of change of position over a time interval:

$v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$

On a position-time graph, average velocity equals the slope of the secant line (the straight line connecting two points on the graph).

To calculate it from a graph:

1. Pick two points on the graph line: $(t_1, x_1)$ and $(t_2, x_2)$

2. Find the change in position: $\Delta x = x_2 - x_1$

3. Find the change in time: $\Delta t = t_2 - t_1$

4. Divide: $v_{avg} = \frac{\Delta x}{\Delta t}$

For example, if an object is at $x = 2 \text{ m}$ at $t = 1 \text{ s}$ and at $x = 10 \text{ m}$ at $t = 5 \text{ s}$, then $v_{avg} = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2 \text{ m/s}$.

Note that average velocity can be positive, negative, or zero depending on the direction of the displacement. If the object ends up back where it started, $\Delta x = 0$ and the average velocity is zero, even if the object moved the entire time.

Instantaneous Velocity Using Tangent Lines

Instantaneous velocity is the velocity at one specific moment in time. It's what a speedometer reads (though a speedometer shows speed, not velocity, since it doesn't indicate direction).

On a position-time graph, instantaneous velocity equals the slope of the tangent line at that point. A tangent line is a straight line that just touches the curve at a single point and matches the curve's direction there.

To find instantaneous velocity from a curved position-time graph:

1. Identify the point on the curve at the time you care about.
2. Draw a tangent line that touches the curve only at that point, following the curve's direction.
3. Pick two convenient points on the tangent line (not on the curve): $(t_1, x_1)$ and $(t_2, x_2)$. Choose points that are far apart so your reading is more accurate.
4. Calculate the slope: $v_{inst} = \frac{x_2 - x_1}{t_2 - t_1}$

For a straight-line segment of a position-time graph, the instantaneous velocity is the same everywhere along that segment and equals the average velocity. You only need tangent lines when the graph is curved.

The sign of the tangent line's slope tells you the direction of motion at that instant: positive slope means moving in the positive direction, negative slope means moving in the negative direction, and a slope of zero means the object is momentarily at rest (which often happens at a turning point).

Kinematics and Motion Analysis

Position-time graphs connect directly to the rest of kinematics. The slope of the position-time graph gives you velocity, so if you can read the graph carefully, you can reconstruct how an object's velocity changes over time. This is the foundation for building velocity-time graphs, which you'll use to analyze acceleration next.