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

Step 1: Write the relation between acceleration, velocity, and distance.
Acceleration can be written as \[ a = v \frac{dv}{dx} \] This relation is useful when velocity is given as a function of distance.
Step 2: Analyze the given \(v\)-\(x\) graph.
From the graph, velocity decreases linearly with distance. Hence, \[ v = mx + c \] where \(m\) is a negative constant (since the slope is negative). Therefore, \[ \frac{dv}{dx} = m = \text{constant (negative)} \]
Step 3: Determine the nature of acceleration.
Using \[ a = v \frac{dv}{dx} \] Since \(v\) decreases linearly with \(x\) and \(\frac{dv}{dx}\) is constant, acceleration varies linearly with distance \(x\). Also, because \(\frac{dv}{dx}\) is negative, acceleration increases linearly from a negative value toward zero as \(x\) increases.
Step 4: Match with the given options.
The acceleration–distance graph that shows a straight line with positive slope starting from a negative value corresponds to Graph 1.