AIIMS 2003 Physics Motion with Constant Acceleration MCQ Question

To analyze the variation of speed $v$ with distance $s$ for a body starting from rest and moving with constant acceleration, we begin with the fundamental equations of motion.

Explanation of Correct Answer (Option B)

For an object starting from rest under constant acceleration $a$, the relationship between speed $v$, distance $s$, and acceleration can be derived from the second equation of motion:

$$v^2 = u^2 + 2as$$

Here, $u$ is the initial velocity. Since the body starts from rest, $u = 0$, simplifying the equation to:

$$v^2 = 2as$$

From this equation, we can express $v$ in terms of $s$:

$$v = \sqrt{2as}$$

This relationship indicates that speed $v$ increases with the square root of the distance $s$ when the acceleration $a$ is constant.

• Graphical Interpretation: The graph of $v$ versus $s$ will be a curve that starts at the origin (0,0) and rises, reflecting the square root relationship. Thus, Option B, which likely represents this curve, is indeed the correct answer.

Why Other Options Are Incorrect

1. Option A - Linear Graph: If this option represents a straight line, it implies a direct linear relationship between speed and distance, described by $v = ks$ (where $k$ is a constant). This would suggest constant speed rather than constant acceleration, which contradicts the premise of the body starting from rest and accelerating.

2. Option C - Parabolic/Quadratic Graph: If this option represents a parabolic curve that opens upwards in a manner akin to $v = as^2 + b$, this is not consistent with the kinematic equation we derived. The motion is not quadratic in terms of distance since the relationship involves the square root of the distance, not the square of it.

3. Option D - Exponential Graph: If this option depicts an exponential curve, it suggests a rapid increase in speed that does not align with the constant acceleration model. Under constant acceleration, the increase in speed is not exponential but follows the square root function.

Conclusion

In summary, the correct representation of the variation of speed $v$ with distance $s$ for an object starting from rest and moving with constant acceleration is a curve that reflects the square root relationship, as given by $v = \sqrt{2as}$. Thus, Option B is the correct answer while the other options misrepresent the relationship due to their assumptions of linearity, quadratic, or exponential growth.