Motion Graphs

Introduction

Graphs are powerful tools for visualizing and analyzing motion. The three main types are:

1. Position-time graphs ($x$ vs $t$)
2. Velocity-time graphs ($v$ vs $t$)
3. Acceleration-time graphs ($a$ vs $t$)

Position-Time Graphs

Interpreting Slope

The slope of a position-time graph gives the velocity.

$\text{slope} = \frac{\Delta x}{\Delta t} = v$

Key features:

• Horizontal line → zero velocity (at rest)
• Positive slope → positive velocity (moving forward)
• Negative slope → negative velocity (moving backward)
• Steeper slope → greater speed
• Curved line → changing velocity (acceleration)

Interpreting Curvature

• Straight line: constant velocity ($a = 0$)
• Curve upward: increasing velocity (positive acceleration)
• Curve downward: decreasing velocity (negative acceleration)

Reading the Graph

• y-value: position at that time
• Slope: velocity at that time
• Change in slope: acceleration

Velocity-Time Graphs

Interpreting Slope

The slope of a velocity-time graph gives the acceleration.

$\text{slope} = \frac{\Delta v}{\Delta t} = a$

Key features:

• Horizontal line → zero acceleration (constant velocity)
• Positive slope → positive acceleration
• Negative slope → negative acceleration (deceleration)
• Steeper slope → greater acceleration

Interpreting Area

The area under a velocity-time graph gives the displacement.

$\text{area} = \int v \, dt = \Delta x$

For constant velocity (rectangle): $\Delta x = v \cdot t$

For constant acceleration (trapezoid or triangle): $\Delta x = \frac{1}{2}(v_0 + v)t$

Sign matters:

• Area above time axis → positive displacement
• Area below time axis → negative displacement

Reading the Graph

• y-value: velocity at that time
• Slope: acceleration at that time
• Area under curve: displacement

Acceleration-Time Graphs

Interpreting Area

The area under an acceleration-time graph gives the change in velocity.

$\text{area} = \int a \, dt = \Delta v$

For constant acceleration: $\Delta v = a \cdot t$

Reading the Graph

• y-value: acceleration at that time
• Area under curve: change in velocity
• Horizontal line: constant acceleration

Relationships Between Graphs

| If you have... | To get... | Operation | |----------------|-----------|-----------| | Position-time | Velocity-time | Take slope (derivative) | | Velocity-time | Acceleration-time | Take slope (derivative) | | Velocity-time | Position-time | Find area (integral) | | Acceleration-time | Velocity-time | Find area (integral) |

Common Motion Patterns

Constant Velocity

• Position-time: straight line (slope = velocity)
• Velocity-time: horizontal line
• Acceleration-time: zero (horizontal at $a = 0$)

Constant Acceleration

• Position-time: parabola (curved)
• Velocity-time: straight line (slope = acceleration)
• Acceleration-time: horizontal line

Speeding Up vs. Slowing Down

Speeding up: velocity and acceleration have the same sign

• Moving right and accelerating right: $v > 0$, $a > 0$
• Moving left and accelerating left: $v < 0$, $a < 0$

Slowing down: velocity and acceleration have opposite signs

• Moving right and accelerating left: $v > 0$, $a < 0$
• Moving left and accelerating right: $v < 0$, $a > 0$

Problem-Solving Tips

1. Identify the type of graph (position, velocity, or acceleration)
2. Look at the y-value for the quantity itself
3. Calculate slope to find the derivative quantity
4. Calculate area to find the integral quantity
5. Check signs carefully (positive/negative)
6. Draw the graph if only given data