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Two objects of masses 1 × 10–3 kg and 4 × 10–3 kg have equal momentum. What is the ratio of their kinetic energies
- a)4 : 1
- b)2 : 1
- c)16 : 1
- d)2 :1
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
Two objects of masses 1 × 10–3 kg and 4 × 10–3 kg have equal momentum...
Given:
Mass of object 1, m1 = 1 × 10^(-3) kg
Mass of object 2, m2 = 4 × 10^(-3) kg
Momentum of object 1, p1 = Momentum of object 2, p2
To find:
The ratio of their kinetic energies
Solution:
The momentum of an object is given by the product of its mass and velocity:
p = mv
Step 1: Calculate the velocities of the two objects using their momenta and masses:
For object 1:
p1 = m1v1
v1 = p1 / m1
For object 2:
p2 = m2v2
v2 = p2 / m2
Step 2: Calculate the kinetic energies of the two objects using their masses and velocities:
The kinetic energy of an object is given by the formula:
KE = (1/2)mv^2
For object 1:
KE1 = (1/2)m1v1^2
For object 2:
KE2 = (1/2)m2v2^2
Step 3: Calculate the ratio of their kinetic energies:
Ratio = KE1 / KE2
= [(1/2)m1v1^2] / [(1/2)m2v2^2]
= (m1/m2)(v1/v2)^2
Substituting the values of v1 and v2 from Step 1:
Ratio = (m1/m2)((p1/m1)/(p2/m2))^2
= (m1/m2)(p1^2 / p2^2)
= (m1p1^2) / (m2p2^2)
Since p1 = p2 (given in the question), substituting the value of p1:
Ratio = (m1p1^2) / (m2p1^2)
= m1 / m2
Step 4: Substitute the given values of m1 and m2:
Ratio = (1 × 10^(-3) kg) / (4 × 10^(-3) kg)
= 1/4
Simplifying the ratio, we get:
Ratio = 1 : 4
Therefore, the ratio of their kinetic energies is 1 : 4, which corresponds to option A.
Mass of object 1, m1 = 1 × 10^(-3) kg
Mass of object 2, m2 = 4 × 10^(-3) kg
Momentum of object 1, p1 = Momentum of object 2, p2
To find:
The ratio of their kinetic energies
Solution:
The momentum of an object is given by the product of its mass and velocity:
p = mv
Step 1: Calculate the velocities of the two objects using their momenta and masses:
For object 1:
p1 = m1v1
v1 = p1 / m1
For object 2:
p2 = m2v2
v2 = p2 / m2
Step 2: Calculate the kinetic energies of the two objects using their masses and velocities:
The kinetic energy of an object is given by the formula:
KE = (1/2)mv^2
For object 1:
KE1 = (1/2)m1v1^2
For object 2:
KE2 = (1/2)m2v2^2
Step 3: Calculate the ratio of their kinetic energies:
Ratio = KE1 / KE2
= [(1/2)m1v1^2] / [(1/2)m2v2^2]
= (m1/m2)(v1/v2)^2
Substituting the values of v1 and v2 from Step 1:
Ratio = (m1/m2)((p1/m1)/(p2/m2))^2
= (m1/m2)(p1^2 / p2^2)
= (m1p1^2) / (m2p2^2)
Since p1 = p2 (given in the question), substituting the value of p1:
Ratio = (m1p1^2) / (m2p1^2)
= m1 / m2
Step 4: Substitute the given values of m1 and m2:
Ratio = (1 × 10^(-3) kg) / (4 × 10^(-3) kg)
= 1/4
Simplifying the ratio, we get:
Ratio = 1 : 4
Therefore, the ratio of their kinetic energies is 1 : 4, which corresponds to option A.
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Community Answer
Two objects of masses 1 × 10–3 kg and 4 × 10–3 kg have equal momentum...
K.E. = ½ mv2
p= mv => K.E. = ½ m (p/m)2 = p2/2m
∴ K.E1/K.E2 = (P12/2m1)/(p22/2m2) = m2/m1 = p12/p22
Here 'p1' = p2 (given)
∴ K.E1/K.E2 = 4×10-3 kg/ 1×10-3 kg = 4/1
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Two objects of masses 1 × 10–3 kg and 4 × 10–3 kg have equal momentum. What is the ratio of their kinetic energiesa)4 : 1b)2 : 1c)16 : 1d)2 :1Correct answer is option 'A'. Can you explain this answer?
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