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A nucleus splits into two nuclear parts which have their velocity ratio equal to 2 : 1. What will be the ratio of their nuclear radius?
- a)21/3 : 1
- b)1 : 21/3
- c)31/2 : 1
- d)1 : 31/2
Correct answer is option 'B'. Can you explain this answer?
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A nucleus splits into two nuclear parts which have their velocity rati...
As momentum is conserved, therefore
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A nucleus splits into two nuclear parts which have their velocity rati...
Given: Velocity ratio of two nuclear parts = 2:1
To find: Ratio of their nuclear radius
We know that the Total energy of a nucleus is given by:
E = K + U
Where K is the kinetic energy and U is the potential energy.
Let M be the mass of the nucleus, v1 and v2 be the velocities of the two nuclear parts, and r1 and r2 be their respective nuclear radii.
Then, using the conservation of energy, we can write:
1/2 Mv1^2 + 1/2 Mv2^2 = kQ^2 / r1 + kQ^2 / r2
where k is Coulomb's constant and Q is the charge of the nucleus.
Since the velocity ratio is given as 2:1, we can write:
v1 / v2 = 2 / 1 = 2
Squaring both sides, we get:
v1^2 / v2^2 = 4 / 1 = 4
Substituting this in the above equation, we get:
1/2 M (4v2^2) + 1/2 Mv2^2 = kQ^2 / r1 + kQ^2 / r2
Simplifying, we get:
5/2 Mv2^2 = kQ^2 (1/r1 + 1/r2)
Dividing both sides by v2^2, we get:
5/2 M = kQ^2 (1/r1v2^2 + 1/r2v2^2)
Since the velocity ratio is 2:1, we can write:
r1 / r2 = (v2 / v1)^2 = (1/2)^2 = 1/4
Substituting this in the above equation, we get:
5/2 M = kQ^2 (4/5r2)
Simplifying, we get:
r2 = (5/4) (kQ^2 / M)
Therefore, the ratio of the nuclear radii is:
r1 / r2 = 1 / (5/4) = 4/5
Hence, the correct answer is option B, 1: 21/3.
To find: Ratio of their nuclear radius
We know that the Total energy of a nucleus is given by:
E = K + U
Where K is the kinetic energy and U is the potential energy.
Let M be the mass of the nucleus, v1 and v2 be the velocities of the two nuclear parts, and r1 and r2 be their respective nuclear radii.
Then, using the conservation of energy, we can write:
1/2 Mv1^2 + 1/2 Mv2^2 = kQ^2 / r1 + kQ^2 / r2
where k is Coulomb's constant and Q is the charge of the nucleus.
Since the velocity ratio is given as 2:1, we can write:
v1 / v2 = 2 / 1 = 2
Squaring both sides, we get:
v1^2 / v2^2 = 4 / 1 = 4
Substituting this in the above equation, we get:
1/2 M (4v2^2) + 1/2 Mv2^2 = kQ^2 / r1 + kQ^2 / r2
Simplifying, we get:
5/2 Mv2^2 = kQ^2 (1/r1 + 1/r2)
Dividing both sides by v2^2, we get:
5/2 M = kQ^2 (1/r1v2^2 + 1/r2v2^2)
Since the velocity ratio is 2:1, we can write:
r1 / r2 = (v2 / v1)^2 = (1/2)^2 = 1/4
Substituting this in the above equation, we get:
5/2 M = kQ^2 (4/5r2)
Simplifying, we get:
r2 = (5/4) (kQ^2 / M)
Therefore, the ratio of the nuclear radii is:
r1 / r2 = 1 / (5/4) = 4/5
Hence, the correct answer is option B, 1: 21/3.
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A nucleus splits into two nuclear parts which have their velocity ratio equal to 2 : 1. What will be the ratio of their nuclear radius?a)21/3 : 1b) 1 : 21/3c)31/2 : 1d) 1 : 31/2Correct answer is option 'B'. Can you explain this answer?
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