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Question #43659

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $M$ is moving in a horizontal plane along a circular path of radius $R$ with constant angular speed $\omega$ , constrained by a massless radial string. At a certain instant, the string snaps and simultaneously the particle ejects a fragment of mass $m$ tangentially forward. In the reference frame moving with the instantaneous velocity of the particle just before ejection, the ejected fragment has a speed $u$ in the forward direction. What is the kinetic energy of the remaining fragment in the laboratory frame immediately after the ejection?

Options

A.
$\frac{1}{2} (M-m) \omega^2 R^2 - m u \omega R + \frac{m^2 u^2}{2(M-m)}$
B.
$\frac{1}{2} (M-m) \omega^2 R^2 + m u \omega R + \frac{m^2 u^2}{2(M-m)}$
C.
$\frac{1}{2} (M-m) \omega^2 R^2 + \frac{m^2 u^2}{2(M-m)}$
D.
$\frac{1}{2} M \omega^2 R^2 + \frac{m^2 u^2}{2(M-m)}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Recoil of Atom|Relative Velocity in 2D|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between vector momentum conservation and scalar energy adjustments in non-zero momentum frames.

A common lethal trap for JEE Advanced students is calculating the recoil energy in the center-of-mass frame ( $E_{recoil} = \frac{m^2 u^2}{2(M-m)}$ ) and erroneously adding it directly to the base kinetic energy of the remaining mass (Option C).

This scalar approach completely ignores the Galilean transformation of kinetic energy, specifically the cross-term ( $-mu\omega R$ ) that arises from the dot product of the initial velocity vector and the recoil velocity vector.

By forcing the emission to be tangential, the cross-term is maximized, heavily penalizing students who bypass strict vector momentum conservation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Recoil of Atom
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free