2.8 Graphical Analysis of One-Dimensional Motion

Graphical Analysis of One-Dimensional Motion

Graphs are one of the most powerful tools for understanding motion. Instead of just plugging numbers into equations, you can see what an object is doing: speeding up, slowing down, changing direction, or standing still. Position-time, velocity-time, and acceleration-time graphs are all connected to each other through slopes and areas, and learning to read them fluently will make the rest of kinematics much easier.

Graphical Analysis of Position, Velocity, and Acceleration

Interpretation of straight-line graphs

Every straight-line graph in kinematics gives you two key pieces of information: the slope and the y-intercept.

The slope tells you the rate of change of whatever is on the y-axis with respect to whatever is on the x-axis.

• On a position-time graph, the slope is the velocity. A line rising steeply means the object is moving fast; a flat line means it's at rest.
• On a velocity-time graph, the slope is the acceleration. A line tilting upward means the object is speeding up (in the positive direction); a line tilting downward means it's slowing down or accelerating in the negative direction.

The y-intercept tells you the starting value of the y-axis quantity (at $t = 0$).

• On a position-time graph, the y-intercept is the initial position $x_0$. If a runner lines up 5 m ahead of the starting line, $x_0 = 5 \text{ m}$.
• On a velocity-time graph, the y-intercept is the initial velocity $v_0$. If a ball is thrown upward at 12 m/s, $v_0 = 12 \text{ m/s}$.

Velocity from position-time graphs

Average velocity is the slope of the straight line connecting two points on a position-time graph:

$v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

For example, if a train is at position $x_i = 20 \text{ m}$ at $t_i = 2 \text{ s}$ and at $x_f = 80 \text{ m}$ at $t_f = 6 \text{ s}$, its average velocity is $\frac{80 - 20}{6 - 2} = 15 \text{ m/s}$.

Instantaneous velocity is the velocity at a single moment in time. On a curved position-time graph, you find it by drawing the tangent line at that point and calculating its slope. On a straight-line graph, the instantaneous velocity is the same everywhere because the slope never changes.

Acceleration from velocity-time graphs

Average acceleration works the same way, but now you're looking at a velocity-time graph:

$a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t_f - t_i}$

If a plane's velocity increases from $v_i = 0 \text{ m/s}$ to $v_f = 80 \text{ m/s}$ over $\Delta t = 20 \text{ s}$ during takeoff, its average acceleration is $\frac{80 - 0}{20} = 4 \text{ m/s}^2$.

Instantaneous acceleration is found by drawing the tangent line to the velocity-time curve at a specific instant and calculating its slope. For a straight-line velocity-time graph, the acceleration is constant, so the instantaneous and average values are the same.

Construction of velocity-time graphs from position-time graphs

Since velocity is the slope of the position-time graph, you can build a velocity-time graph step by step:

1. Pick several points (or segments) on the position-time graph.
2. At each point, calculate the slope. For straight segments, use $\frac{\Delta x}{\Delta t}$. For curves, draw a tangent line and find its slope.
3. Plot each slope value on a new graph with velocity on the y-axis and time on the x-axis.

A constant-slope segment on the position-time graph becomes a horizontal line on the velocity-time graph. A segment that curves upward (getting steeper) becomes a rising line on the velocity-time graph, indicating positive acceleration.

Creation of acceleration-time graphs from velocity-time graphs

The same logic applies one level deeper. Acceleration is the slope of the velocity-time graph:

1. Pick several points or segments on the velocity-time graph.
2. Calculate the slope at each point using $\frac{\Delta v}{\Delta t}$ or a tangent line.
3. Plot those values on a new graph with acceleration on the y-axis and time on the x-axis.

For uniform (constant) acceleration, the velocity-time graph is a straight line, and the acceleration-time graph is just a horizontal line at that constant value.

Relationships Between Position, Velocity, and Acceleration Graphs

Connections between position-time, velocity-time, and acceleration-time graphs

These three graphs are linked by two operations: slopes (going from left to right in the list below) and areas (going from right to left).

• Position-time graph:
  • Slope at any point gives velocity at that time.
  • A steeper slope means a larger velocity. A negative slope means the object is moving in the negative direction.
  • A straight line means constant velocity. A curve means the velocity is changing (the object is accelerating).
• Velocity-time graph:
  • Slope at any point gives acceleration at that time.
  • A positive slope means velocity is increasing; a negative slope means velocity is decreasing.
  • Area under the curve between two times gives the displacement (change in position) during that interval. Areas above the time axis count as positive displacement; areas below count as negative.
• Acceleration-time graph:
  • Area under the curve between two times gives the change in velocity during that interval.

Fundamental Concepts in Motion Analysis

A few definitions that come up constantly when working with motion graphs:

• Kinematics is the branch of physics that describes motion without worrying about what forces cause it. Graphs are a core tool of kinematics.
• Vector quantities have both magnitude and direction. Velocity and acceleration are vectors, which is why they can be positive or negative on a graph.
• Scalar quantities have magnitude only. Speed (the absolute value of velocity) and distance are scalars.
• Reference frame is the coordinate system you choose to describe an object's position and motion. Your choice of origin and positive direction affects the signs on your graphs.