A stationary bomb explodes into two parts, \(4\text{ kg}\) and \(8\text{ kg}\). The velocity of the \(8\text{ kg}\) mass is \(6\text{ ms}^{-1}\). The KE of the other body is:
1. \(48\text{ J}\)
2. \(24\text{ J}\)
3. \(288\text{ J}\)
4. \(16\text{ J}\)
View Answer
By conservation of momentum, \(m_1 v_1 = m_2 v_2\), which gives \(4 \times v_1 = 8 \times 6\), so \(v_1 = 12\text{ ms}^{-1}\). The kinetic energy of the \(4\text{ kg}\) body is \(K = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \times 4 \times 12^2 = 288\text{ J}\).
A stationary object explodes into two parts of equal masses, then:
Assertion: Both parts will have same kinetic energy after explosion.
Reason : Both parts will have same momentum after explosion.
1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Assertion and Reason are false.
View Answer
By conservation of momentum, the two parts move in opposite directions with equal magnitude of momentum, \( \vec{p}_1 = -\vec{p}_2 \). Thus, they have different momenta (since momentum is a vector), so the Reason is false. However, since they have the same mass and same momentum magnitude, their kinetic energy \( K = \frac{p^2}{2m} \) is the same, so the Assertion is true.
Assertion: If kinetic energy of a system of particles is zero, then linear momentum of system must be zero.
Reason: If linear momentum of a system of particles is zero, then kinetic energy of system must be zero.
1. Both Assertion and Reason are true and Reason is the correct explanation of Assertion.
2. Both Assertion and Reason are true but Reason is not correct explanation of Assertion.
3. Assertion is true but Reason is false.
4. Assertion and Reason are false.
View Answer
If the kinetic energy of a system is zero, the speed of each particle must be zero, meaning the total linear momentum is also zero. If the total linear momentum is zero, particles can still be moving in opposite directions, resulting in a non-zero kinetic energy. Thus, the Assertion is true but the Reason is false.
A bullet of mass \( m \) leaves the barrel of a gun of mass \( M \) with a velocity \( v \). The gun is known to recoil with a velocity \( V \). If \( k \) and \( K \) respectively denote the kinetic energies of the bullet and the gun respectively; then
1. \( K = \left(\frac{m}{M}\right)^2 k \)
2. \( K = \sqrt{\frac{m}{M}} k \)
3. \( K = \left(\frac{m}{M}\right) k \)
4. \( K = \left(\frac{M}{m}\right) k \)
View Answer
By conservation of momentum, the bullet and gun have equal momentum magnitude, \( p \). Since kinetic energy is \( K_{\text{E}} = \frac{p^2}{2\text{mass}} \), we have \( k = \frac{p^2}{2m} \) and \( K = \frac{p^2}{2M} \). Thus, \( K = \left(\frac{m}{M}\right) k \).