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A body of mass 5 kg explodes at rest into three fragments with masses in the ratio 1 : 1 : 3. The fragements with equal masses fly in mutually perpendicular directions with speeds of 21 m/sec. The velocity of the heaviest fragment will be
- a)11.5 m/sec
- b)14.0 m/sec
- c)7.0 m/sec
- d)9.87 m/sec
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A body of mass 5 kg explodes at rest into three fragments with masses ...
Momentum of first body
p1 = 1 x 21 = 21 kg-m / sec.
Momentum of second body
p2 = 1 x 21 = 21 kg. m / sec.
Momentum of third body p3 = 3V kg - m / sec
Initial momentum = zero
∴ final momentum = 0
p1 = 1 x 21 = 21 kg-m / sec.
Momentum of second body
p2 = 1 x 21 = 21 kg. m / sec.
Momentum of third body p3 = 3V kg - m / sec
Initial momentum = zero
∴ final momentum = 0
Most Upvoted Answer
A body of mass 5 kg explodes at rest into three fragments with masses ...
Given:
- Mass of the body (m) = 5 kg
- Mass ratio of the fragments: 1 : 1 : 3
- Speed of the fragments with equal masses (v) = 21 m/s
To find:
Velocity of the heaviest fragment
Solution:
Step 1: Finding the mass of each fragment
Let the masses of the fragments be m1, m2, and m3.
Given that the mass ratio is 1 : 1 : 3, we can write the equation:
m1 : m2 : m3 = 1 : 1 : 3
Let the common ratio be x, so we can rewrite the equation as:
m1 = x
m2 = x
m3 = 3x
Since the sum of the masses of the fragments must equal the mass of the body, we have:
m1 + m2 + m3 = 5
Substituting the values of m1, m2, and m3, we get:
x + x + 3x = 5
5x = 5
x = 1
Therefore, the masses of the fragments are:
m1 = 1 kg
m2 = 1 kg
m3 = 3 kg
Step 2: Finding the velocity of the heaviest fragment
The fragments with equal masses are flying in mutually perpendicular directions. This means that they form a right-angled triangle, and we can apply the Pythagorean theorem to find the magnitude of their resultant velocity.
Let the magnitude of the resultant velocity be V.
According to the Pythagorean theorem:
V^2 = (v1)^2 + (v2)^2
V^2 = (v)^2 + (v)^2
V^2 = 2v^2
V = sqrt(2v^2)
V = sqrt(2) * v
Since the mass of the third fragment is 3 kg, its velocity will be proportional to the square root of its mass. Therefore, the velocity of the heaviest fragment (V3) will be:
V3 = sqrt(3) * V
Substituting the value of V from above, we get:
V3 = sqrt(3) * sqrt(2) * v
V3 = sqrt(6) * v
Given that v = 21 m/s, we can calculate the velocity of the heaviest fragment:
V3 = sqrt(6) * 21
V3 ≈ 9.87 m/s
Therefore, the correct answer is option D) 9.87 m/s.
- Mass of the body (m) = 5 kg
- Mass ratio of the fragments: 1 : 1 : 3
- Speed of the fragments with equal masses (v) = 21 m/s
To find:
Velocity of the heaviest fragment
Solution:
Step 1: Finding the mass of each fragment
Let the masses of the fragments be m1, m2, and m3.
Given that the mass ratio is 1 : 1 : 3, we can write the equation:
m1 : m2 : m3 = 1 : 1 : 3
Let the common ratio be x, so we can rewrite the equation as:
m1 = x
m2 = x
m3 = 3x
Since the sum of the masses of the fragments must equal the mass of the body, we have:
m1 + m2 + m3 = 5
Substituting the values of m1, m2, and m3, we get:
x + x + 3x = 5
5x = 5
x = 1
Therefore, the masses of the fragments are:
m1 = 1 kg
m2 = 1 kg
m3 = 3 kg
Step 2: Finding the velocity of the heaviest fragment
The fragments with equal masses are flying in mutually perpendicular directions. This means that they form a right-angled triangle, and we can apply the Pythagorean theorem to find the magnitude of their resultant velocity.
Let the magnitude of the resultant velocity be V.
According to the Pythagorean theorem:
V^2 = (v1)^2 + (v2)^2
V^2 = (v)^2 + (v)^2
V^2 = 2v^2
V = sqrt(2v^2)
V = sqrt(2) * v
Since the mass of the third fragment is 3 kg, its velocity will be proportional to the square root of its mass. Therefore, the velocity of the heaviest fragment (V3) will be:
V3 = sqrt(3) * V
Substituting the value of V from above, we get:
V3 = sqrt(3) * sqrt(2) * v
V3 = sqrt(6) * v
Given that v = 21 m/s, we can calculate the velocity of the heaviest fragment:
V3 = sqrt(6) * 21
V3 ≈ 9.87 m/s
Therefore, the correct answer is option D) 9.87 m/s.
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A body of mass 5 kg explodes at rest into three fragments with masses in the ratio 1 : 1 : 3. The fragements with equal masses fly in mutually perpendicular directions with speeds of 21 m/sec. The velocity of the heaviest fragment will bea)11.5 m/secb)14.0 m/secc)7.0 m/secd)9.87 m/secCorrect answer is option 'D'. Can you explain this answer?
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