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

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

Since velocity is constant, the area is a rectangle: $\text{Area} = \text{base} \times \text{height}$ $\Delta x = t \times v = 8 \times 15 = 120 \text{ m}$

Part 2: Acceleration Acceleration = slope of velocity-time graph

Since the line is horizontal: $a = \frac{\Delta v}{\Delta t} = \frac{0}{8} = 0 \text{ m/s}^2$

Answers:

• Displacement: 120 m
• Acceleration: 0 m/s² (moving at constant velocity)

Key insight: Horizontal line on $v$-$t$ graph means constant velocity (zero acceleration).