Velocity–time graphs Knowledge-Based Study

Study notes

1. Understanding the Axes

   A velocity–time ($v$–$t$) graph illustrates how the velocity of an object changes over time. In standard kinematic analysis:

   • Horizontal Axis ($t$): Represents elapsed time, typically measured in seconds (s).
   • Vertical Axis ($v$): Represents instantaneous velocity, typically measured in meters per second (m/s).

   Because velocity is a vector quantity, values on the vertical axis can be positive or negative, corresponding to the direction of motion relative to a chosen reference frame.

   Key Features of the Graph

   A flat line located directly on the time axis indicates that velocity is zero ($v = 0$), meaning the object is at rest. A horizontal line above or below the time axis indicates that the object is traveling at a constant velocity with zero acceleration.

   The Meaning of Slope

   The gradient (slope) of a line on a velocity–time graph represents the object's acceleration.

   Slope is defined as the change in the vertical coordinate divided by the change in the horizontal coordinate:

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

   Interpreting Different Slopes

   • Positive Slope: The velocity is increasing in the positive direction or decreasing in the negative direction, indicating positive acceleration.
   • Zero Slope: The line is horizontal, meaning velocity does not change over time ($a = 0$).
   • Negative Slope: The velocity is decreasing in the positive direction or increasing in the negative direction, indicating negative acceleration.
   • Straight Line vs. Curved Line: A straight line indicates constant (uniform) acceleration, whereas a curved line indicates changing (non-uniform) acceleration.

   Finding Displacement

   The area bounded between the graph line and the time axis represents the object's displacement ($\Delta x$).

   Because velocity is defined as $\Delta x / \Delta t$, the product of velocity and time yields displacement:

   $$\Delta x = v \times \Delta t$$

   Calculating Geometric Shapes

   For uniform acceleration, graphs form standard geometric shapes that can be decomposed:

   • Rectangles: Used for regions of constant velocity, where $\text{Area} = \text{base} \times \text{height}$.
   • Triangles: Used for regions where an object starts from rest or comes to a stop with constant acceleration, where $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
   • Trapezoids: Used for regions transitioning between two non-zero velocities.

   Sign Convention: Area located above the time axis represents positive displacement, whereas area located below the time axis represents negative displacement. To find total displacement, sum the areas taking their signs into account. To find total distance traveled, sum the absolute values of all areas.

   Reversing Direction vs. Slowing Down

   A frequent error is assuming an object reverses direction when the slope becomes negative. However:

   • A negative slope simply means negative acceleration. If the line is still above the time axis ($v > 0$), the object is still moving forward; it is simply slowing down.
   • An object changes its direction of motion only when the graph crosses the time axis—that is, when velocity changes sign from positive to negative or vice versa.

   Distinguishing $v$–$t$ Graphs from $x$–$t$ Graphs

   Avoid confusing position–time ($x$–$t$) graphs with velocity–time graphs:

   • A horizontal line on an $x$–$t$ graph means the object is at rest, whereas on a $v$–$t$ graph it represents steady motion at constant speed.
   • The slope of an $x$–$t$ graph gives velocity, whereas the slope of a $v$–$t$ graph gives acceleration.