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Atoms and Nuclei question

2018 · 15 Apr · Shift 2 · Q70
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Atoms and Nuclei question

2018 · 15 Apr · Shift 2 · Q70

JEE MainPhysicsAtoms and NucleiMCQ+4 / −1
An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of 8 : 27. The ratio of the radii of the nuclei (assumed to be spherical) is :
  1. A
    8 : 27
  2. B
    4 : 9
  3. C
    3 : 2
  4. D
    2 : 3
View written solutionFree

Correct answer: C

  1. Use conservation of momentum

Since the भारी nucleus is initially at rest, total initial momentum is zero. After decay into two nuclei,

m1v1=m2v2m_1 v_1 = m_2 v_2m1​v1​=m2​v2​

Given

v1:v2=8:27v_1 : v_2 = 8 : 27v1​:v2​=8:27

So,

m1m2=v2v1=278\frac{m_1}{m_2} = \frac{v_2}{v_1} = \frac{27}{8}m2​m1​​=v1​v2​​=827​

Hence,

m1:m2=27:8m_1 : m_2 = 27 : 8m1​:m2​=27:8

  1. Relate mass to radius

For nuclei, density is approximately constant, so mass is proportional to volume:

m∝R3m \propto R^3m∝R3

Therefore,

m1m2=(R1R2)3\frac{m_1}{m_2} = \left(\frac{R_1}{R_2}\right)^3m2​m1​​=(R2​R1​​)3

Substitute the mass ratio:

(R1R2)3=278\left(\frac{R_1}{R_2}\right)^3 = \frac{27}{8}(R2​R1​​)3=827​

Taking cube root,

R1R2=32\frac{R_1}{R_2} = \frac{3}{2}R2​R1​​=23​

So the ratio of radii is

R1:R2=3:2R_1 : R_2 = 3 : 2R1​:R2​=3:2

  1. Check options
  • A: 8:278:278:27 ❌
  • B: 4:94:94:9 ❌
  • C: 3:23:23:2 ✅
  • D: 2:32:32:3 ❌

Therefore, the correct option is C.

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