Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track as shown in the figure below.
Which of the following statements is correct?
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Speed depends only on vertical height ($h$), but time depends on the path's steepness.
Both the stones reach the bottom at the same time and the same speed.
Both the stones reach the bottom with the same speed and stone I reaches the bottom earlier than stone II.
Both the stones reach the bottom with the same speed and stone II reaches the bottom earlier than stone I.
Both the stones reach the bottom at different times and with different speeds.
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The Correct Option isC
Solution and Explanation
Step 1: Concept
According to the law of conservation of energy, the potential energy at height $h$ ($mgh$) is converted into kinetic energy ($\frac{1}{2}mv^2$) at the bottom.
Step 2: Speed Analysis
Since $mgh = \frac{1}{2}mv^2$, the final speed $v = \sqrt{2gh}$ depends only on the height $h$, not the incline angle. Thus, both stones have the same speed at the bottom.
Step 3: Time Analysis
The acceleration along an incline is $a = g \sin \theta$. The steeper track (Track II) has a larger angle $\theta$, resulting in higher acceleration.
Step 4: Conclusion
Higher acceleration over a shorter path (steep track) means Stone II reaches the bottom first.
Final Answer: (C)
