Understanding Velocity-Time Graphs for Uniform Acceleration

The question asks about the nature of the velocity-time graph for an object moving with uniform acceleration. We are given the equation of motion:

\begin{equation*} v = u + at \end{equation*}

Here, \(v\) is the final velocity at time \(t\), \(u\) is the initial velocity, and \(a\) is the uniform acceleration.

Relating the Equation to Linear Graphs

We are told that velocity (\(v\)) is plotted along the y-axis and time (\(t\)) is plotted along the x-axis. Let's compare the given equation of motion with the general equation of a straight line:

\begin{equation*} y = mx + c \end{equation*}

In this standard linear equation:

• \(y\) represents the variable on the y-axis.
• \(x\) represents the variable on the x-axis.
• \(m\) is the slope of the line.
• \(c\) is the y-intercept (the value of \(y\) when \(x=0\)).

Now, let's map our velocity-time equation \(v = u + at\) to the standard linear equation \(y = mx + c\):

• The variable on the y-axis is velocity, \(v\). So, \(y\) corresponds to \(v\).
• The variable on the x-axis is time, \(t\). So, \(x\) corresponds to \(t\).
• Comparing \(v = u + at\) with \(y = mx + c\), we can see that the term multiplying \(t\) is \(a\), which corresponds to the slope \(m\). Thus, the slope of the velocity-time graph is \(a\) (the acceleration).
• The constant term in the equation \(v = u + at\) is \(u\), which corresponds to the y-intercept \(c\). The y-intercept is the value of \(v\) when \(t=0\). When \(t=0\), the equation becomes \(v = u + a(0) = u\). This confirms that the y-intercept is the initial velocity \(u\).

Since the equation \(v = u + at\) is a linear equation in the form \(y = mx + c\) when \(v\) is on the y-axis and \(t\) is on the x-axis, the graph of velocity versus time will be a straight line.

Analyzing the Options for the Velocity-Time Graph

Let's evaluate the given options based on our findings:

• Option 1: Passing through the origin
  A graph passing through the origin means the y-intercept is zero. The y-intercept of our graph is \(u\). The graph passes through the origin only if the initial velocity \(u\) is zero. This is not always the case for an object moving with uniform acceleration.
• Option 2: With x-intercept u
  The x-intercept is the value of \(t\) when the velocity \(v\) is zero. Setting \(v=0\) in the equation \(0 = u + at\), we get \(t = -u/a\). The x-intercept is \(-u/a\), not necessarily \(u\). Also, time \(t\) must be positive, so a positive x-intercept only occurs if \(u\) and \(a\) have opposite signs (e.g., deceleration).
• Option 3: With y-intercept u
  The y-intercept is the value of the variable on the y-axis (\(v\)) when the variable on the x-axis (\(t\)) is zero. We found that when \(t=0\), \(v = u\). Thus, the y-intercept is indeed the initial velocity \(u\). This option correctly describes the graph.
• Option 4: With slope u
  The slope of the velocity-time graph represents the acceleration. From our analysis, the slope is \(a\), the uniform acceleration, not the initial velocity \(u\). This option is incorrect.

Based on the analysis, the velocity-time graph for an object moving with uniform acceleration, described by \(v = u + at\), is a straight line with a y-intercept equal to the initial velocity \(u\).

Revision Table: Uniform Acceleration Graph Properties

Additional Information: Velocity-Time Graph Insights

The velocity-time graph is a powerful tool for analyzing motion with uniform acceleration. Here are some key points:

• The graph is always a straight line for uniform acceleration.
• The slope of the line indicates the acceleration. A positive slope means positive acceleration, a negative slope means negative acceleration (deceleration), and a zero slope means zero acceleration (constant velocity).
• The y-intercept represents the initial velocity of the object.
• The area under the velocity-time graph between two time points gives the displacement of the object during that time interval. For a straight line, this area can be calculated using geometry (e.g., area of a trapezoid or rectangle/triangle).
• The x-intercept (if it exists for \(t > 0\)) indicates the time at which the object momentarily stops before potentially changing direction. This occurs when the velocity becomes zero.

Understanding the relationship between the equation of motion and its graphical representation is fundamental in kinematics. The linear nature of the velocity-time graph for uniform acceleration simplifies many problems involving motion analysis.