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Physics · System Of Particles

JEE Advanced 2019 — Paper 2 — Question 4

A small particle of mass m moving inside a heavy, hollow and straight tube along the tube axis undergoes elastic collision at two ends. The tube has no friction and it is closed at one end by a flat surface while the other end is fitted with a heavy movable flat piston as shown in figure. When the distance of the piston from closed end is L=L0L=L_{0} the particle speed is v=v0v=v_{0}. The piston is moved inward at a very low speed V such that V≪dLLV0\mathrm{V} \ll \frac{\mathrm{dL}}{\mathrm{L}} \mathrm{V}_{0}, where dL is the infinitesimal displacement of the piston. which of the following statement(s) is/are correct ?

Question figure
  1. Option A:

    The particle's kinetic energy increases by a factor of 4 when the piston is moved inward from L0L_{0} to 12 L0\frac{1}{2} \mathrm{~L}_{0}.

    Correct
  2. Option B:

    After each collision with the piston, the particle speed increases by 2 V .

    Correct
  3. Option C:

    If the piston moves inward by dL , the particle speed increases by 2vdLL2 \mathrm{v} \frac{\mathrm{dL}}{\mathrm{L}}

  4. Option D:

    The rate at which the particle strikes the piston is v/L\mathrm{v} / \mathrm{L}

Answer: A, B

Step-by-step solution

The rate of collision of the particle with the piston is 12L/v=v2L\frac{1}{2 L / v}=\frac{v}{2 L}

The speed of the particle after a collision with the piston is :

v+2 V\mathrm{v}+2 \mathrm{~V}, when it collides with it with a speed v .

if the piston moves inward by dL the speed of the particle increases by

dv=2 V×dLV×v2 L\mathrm{dv}=2 \mathrm{~V} \times \frac{\mathrm{dL}}{\mathrm{V}} \times \frac{\mathrm{v}}{2 \mathrm{~L}}

i.e. dvv=dLL\frac{d v}{v}=\frac{d L}{L}

Since kinetic energy, K=12mv2,dKK=2(−dL)LK=\frac{1}{2} \mathrm{mv}^{2}, \frac{\mathrm{dK}}{\mathrm{K}}=\frac{2(-\mathrm{dL})}{\mathrm{L}} (after putting the proper sign)

Integrating, KL2=\mathrm{KL}^{2}= constant.

∴Kf=4 Ki\therefore \mathrm{K}_{\mathrm{f}}=4 \mathrm{~K}_{\mathrm{i}}

Hence correct options are A, B

Solution figure

Answer key and solution verified before publishing.

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Exam
JEE Advanced 2019
Paper
Paper 2
Subject
Physics
Chapter
System Of Particles
Topic
Collisions in One Dimension
A small particle of mass m moving inside a heavy, hollow and straight… | JEE Advanced 2019 PYQ with Solution · DhiX AI