Question

The velocity (\(v\)) – distance (\(x\)) graph is shown in the figure. Which graph represents acceleration (\(a\)) versus distance (\(x\)) variation of this system? 

• A. A
• B. B
• C. C
• D. D

Hint:

When velocity is given as a function of distance, always use \( a = v \frac{dv}{dx} \) to find acceleration.

Answer 1

The Correct Option is A

To find the correct acceleration (\(a\)) versus distance (\(x\)) graph, we need to analyze the given velocity (\(v\)) versus distance (\(x\)) graph.

Step-by-Step Analysis:

1. Understanding the Graph:
   • The velocity (\(v\)) vs. distance (\(x\)) graph shows a straight line with a negative slope.
   • This indicates that velocity decreases linearly with distance.
2. Relation between Velocity, Acceleration, and Distance:
   • From physics, we know that acceleration (\(a\)) is the derivative of velocity with respect to time (\(t\)): \(a = \frac{dv}{dt}\).
   • In terms of distance (\(x\)), acceleration can also be expressed as: \(a = v \frac{dv}{dx}\).
3. Applying the Formula:
   • Given that the velocity (\(v\)) is decreasing linearly with distance (\(x\)), the expression \( \frac{dv}{dx} \) is a constant negative value (slope of the line).
   • Thus, the acceleration \(a = v \frac{dv}{dx}\) also depends on \(v\), which is linearly decreasing. Therefore, acceleration is linearly decreasing and negative, which relates to option A.

Conclusion:

The correct graph for acceleration (\(a\)) versus distance (\(x\)) variation is option A, where the acceleration is linearly decreasing.