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A heavy nucleus at rest breaks into two fragments which fly off with velocities 8:1. The ratio of radii of the fragments is
- a)1:2
- b)1:4
- c)4:1
- d)2:1
Correct answer is option 'A'. Can you explain this answer?
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Explanation:
When a heavy nucleus at rest breaks into two fragments, the conservation of momentum can be applied to determine the velocities of the fragments.
Let the masses of the two fragments be m1 and m2, and their velocities after the breakup be v1 and v2, respectively. According to the conservation of momentum:
Total momentum before the breakup = Total momentum after the breakup
Since the nucleus is at rest initially, the total momentum before the breakup is zero. Therefore, the total momentum after the breakup is also zero.
Mathematically, this can be written as:
0 = m1v1 + m2v2
Given that the ratio of velocities is 8:1, we can write:
v1/v2 = 8/1 = 8
Substituting this ratio into the momentum equation, we have:
0 = m1(8v2) + m2v2
Simplifying the equation:
0 = 8m1v2 + m2v2
0 = v2(8m1 + m2)
Since v2 cannot be zero (as the fragments are flying off), we can conclude that:
8m1 + m2 = 0
Solving this equation, we find:
m1/m2 = -1/8
The negative sign indicates that the fragments have opposite directions of motion. However, the ratio of radii is a positive quantity.
Relation between mass and radius:
The ratio of radii of two objects can be related to the ratio of their masses using the equation:
r1/r2 = (m2/m1)^(1/3)
where r1 and r2 are the radii of the fragments.
Substituting the values of m1/m2 and simplifying, we have:
r1/r2 = (-1/8)^(1/3)
Taking the cube root of (-1/8), we get:
r1/r2 = -1/2
Since the ratio of radii cannot be negative, we take the absolute value, which gives:
|r1/r2| = 1/2
Comparing this with the given options, we can see that the correct answer is option 'A': 1:2, which represents a positive ratio of 1:2.
When a heavy nucleus at rest breaks into two fragments, the conservation of momentum can be applied to determine the velocities of the fragments.
Let the masses of the two fragments be m1 and m2, and their velocities after the breakup be v1 and v2, respectively. According to the conservation of momentum:
Total momentum before the breakup = Total momentum after the breakup
Since the nucleus is at rest initially, the total momentum before the breakup is zero. Therefore, the total momentum after the breakup is also zero.
Mathematically, this can be written as:
0 = m1v1 + m2v2
Given that the ratio of velocities is 8:1, we can write:
v1/v2 = 8/1 = 8
Substituting this ratio into the momentum equation, we have:
0 = m1(8v2) + m2v2
Simplifying the equation:
0 = 8m1v2 + m2v2
0 = v2(8m1 + m2)
Since v2 cannot be zero (as the fragments are flying off), we can conclude that:
8m1 + m2 = 0
Solving this equation, we find:
m1/m2 = -1/8
The negative sign indicates that the fragments have opposite directions of motion. However, the ratio of radii is a positive quantity.
Relation between mass and radius:
The ratio of radii of two objects can be related to the ratio of their masses using the equation:
r1/r2 = (m2/m1)^(1/3)
where r1 and r2 are the radii of the fragments.
Substituting the values of m1/m2 and simplifying, we have:
r1/r2 = (-1/8)^(1/3)
Taking the cube root of (-1/8), we get:
r1/r2 = -1/2
Since the ratio of radii cannot be negative, we take the absolute value, which gives:
|r1/r2| = 1/2
Comparing this with the given options, we can see that the correct answer is option 'A': 1:2, which represents a positive ratio of 1:2.
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