Question

A bomb is 12 kg and divides into two parts whose ratio of masses is 1:3. If the kinetic energy of a smaller part is 216 J, then the momentum of a bigger part in kg-m/sec will be
A. 36
B. 72
C. 108
D. Data is incomplete

Answer 1

Hint:To solve this problem first we have to find the velocity of the bigger part by using the principle of conservation of momentum. Then we can use the formula for kinetic energy and use the values accordingly.






Formula used:
 Momentum is given as,
\[p = m \times v\]
Where m is mass
v is velocity
By law of conservation of momentum,
Total initial momentum of the system is equal to the total final momentum of the system






Complete answer:
Given mass of a bomb, m = 12kg
The ratio of masses of the smaller part of the bomb to the bigger part is, \[{m_A}:{m_B} = 1:3\]
The final kinetic energy of the smaller part, \[K{E_A} = 216J\]
Let the final velocity of the smaller part and the bigger part be \[{v_A}\]and \[{v_B}\].
Initial momentum of a system is zero.
The final momentum of the system is the sum of the momentum of A and B as,
\[{m_A}{v_A} + {m_B}{v_B} = {m_A}({v_A} + 3{v_B})\]
By law of conservation of momentum,
Total initial momentum of the system = Total final momentum of the system
\[{m_A}({v_A} + 3{v_B}) = 0\]
\[{v_B} = - \dfrac{{{v_A}}}{3}\] …. (1)
Here, we get the velocity of the bigger part as one-third of the velocity of the smaller part.
Now it is given that final kinetic energy of the smaller part, \[K{E_A} = 216J\]
\[\dfrac{1}{2}{m_A}{v_A}^2 = 216\]
\[\dfrac{1}{2} \times 3 \times {v_A}^2 = 216\]
\[{v_A}^2 = \dfrac{{2 \times 216}}{3}\]
\[{v_A}^2 = 144\]
\[{v_A} = 12m/s\] …. (2)
Here we get the velocity of mass A is \[{v_A} = 12m/s\]
Now momentum of mass B, \[{m_B}{v_B}\]is
\[{m_B}{v_B} = 9 \times ( - \dfrac{{{v_A}}}{3})\] (Using equation 1)
\[{m_B}{v_B} = - 3{v_A}\]
By using equation (2), we get the magnitude of the momentum as,
\[{m_B}{v_B} = 3 \times 12 = 36{\rm{ kg m/s}}\]
Hence option A is the correct answer




Note: Here we use the concept of conservation of linear momentum. In condition for more than two pieces also, linear momentum would have been conserved. As velocity is a vector quantity, it will depend on direction too. A positive sign indicates bodies are moving in the same direction, whereas a negative sign indicates that they are moving in the opposite direction.