3.12 Velocity-Time Graphs

Introduction

A velocity-time graph is a kinematic tool that plots an object's velocity on the vertical axis against time on the horizontal axis. This graphical representation allows one to determine the object's acceleration and displacement by analyzing specific features of the graph.

Interpreting the Graph

1. Reading Velocity

Instantaneous Velocity: The velocity at any specific moment can be read directly from the vertical axis corresponding to that time.

Direction of Motion:

• A positive velocity (graph above the time axis) indicates motion in the positive direction.
• A negative velocity (graph below the time axis) indicates motion in the negative direction.
• A velocity of zero (graph crossing the time axis) means the object is momentarily at rest.

2. The Slope of the Graph: Acceleration

The slope of the line on a velocity-time graph represents the object's acceleration. This relationship is expressed as:

$\text{Slope}=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t}=a$

Rearranging gives the first equation of motion:

$v_f=v_i+at$

Positive Slope: A line sloping upwards indicates positive acceleration. The object is speeding up if its velocity is positive, or slowing down if its velocity is negative.

Negative Slope: A line sloping downwards indicates negative acceleration. The object is slowing down if its velocity is positive (deceleration), or speeding up if its velocity is negative.

Zero Slope (Horizontal Line): A flat, horizontal line indicates zero acceleration. The object is moving at a constant velocity.

3. The Area Under the Graph: Displacement

The area between the graph line and the time axis represents the object's displacement. For a uniformly accelerated object starting with initial velocity $v_i$ and reaching final velocity $v_f$ in time $t$, the area of the trapezoid gives:

$s=\frac{(v_i+v_f)}{2}\cdot t$

This is the second equation of motion (combined with $v_f=v_i+at$, it also yields $s=v_{i}t+\frac{1}{2}a{t}^2$).

• Area Above the Axis: Represents positive displacement (movement in the positive direction).
• Area Below the Axis: Represents negative displacement (movement in the negative direction).
• Total Displacement: The net area is calculated by subtracting areas below the axis from areas above the axis.
• Total Distance: The sum of the absolute values of all areas (all areas treated as positive).

4. Instantaneous Acceleration from a Curved Graph

For a non-linear (curved) v-t graph, the acceleration is not constant. The instantaneous acceleration at any point is found by drawing a tangent to the curve at that point and calculating its slope:

$a_{\text{inst}}={\lim }_{\Delta t\to 0}\frac{\Delta v}{\Delta t}=\text{slope of tangent at that point}$

Relationship with Other Motion Graphs

Speeding Up vs. Slowing Down

An object is speeding up if its velocity is moving away from the time axis (e.g., a positive velocity becoming more positive, or a negative velocity becoming more negative). It is slowing down if its velocity is moving towards the time axis (approaching zero).

Summary

• A velocity-time graph plots velocity vs. time.
• The slope of the graph equals acceleration: $a=\frac{\Delta v}{\Delta t}$, leading to $v_f=v_i+at$.
• The area under the graph equals displacement: $s=\frac{(v_i+v_f)}{2}t$.
• A horizontal line means constant velocity (zero acceleration).
• A straight, sloped line means constant (uniform) acceleration.
• A curved line means non-uniform (changing) acceleration; use the tangent to find instantaneous acceleration.