A bullet of mass \(m\) hits a block of mass \(M\) elastically. The transfer of energy is the maximum, when
In a one-dimensional elastic collision between a moving body of mass \(m\) and a stationary body of mass \(M\), maximum transfer of kinetic energy occurs when the masses are equal (\(m = M\)).
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:
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.
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.
A particle is dropped on the ground from a height \( 36\text{ m} \) and after striking the ground it rebounds to a height of \( 16\text{ m} \). Then the co-efficient of restitution of the collision with the ground is:
The coefficient of restitution \( e \) for a vertical rebound is given by \( e = \sqrt{\frac{h_{\text{rebound}}}{h_{\text{initial}}}} \). Substituting the given values, \( e = \sqrt{\frac{16}{36}} = \frac{4}{6} = \frac{2}{3} \).
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
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, \( K = \left(\frac{m}{M}\right) k \).
Assertion (A): In any kind of collision, kinetic energy cannot be same throughout.
Reason (R): In elastic collision kinetic energy remains constant throughout.
Assertion (A) is true as kinetic energy is not conserved in all types of collisions (e.g., inelastic collisions). Reason (R) is true as kinetic energy is conserved in elastic collisions. However, (R) does not correctly explain (A).
Assertion (A): In a perfectly inelastic collision there is a limit to the loss of kinetic energy of colliding bodies.
Reason (R): In perfectly inelastic collision, linear momentum of system is conserved.
Both (A) and (R) are true. In a perfectly inelastic collision, momentum is conserved (R), which allows the calculation of the final common velocity (\(v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}\)) and thus the minimum kinetic energy (\(KE_f = \frac{1}{2}(m_1+m_2)v_f^2\)) that must remain, placing a limit on kinetic energy loss (A). Hence, (R) correctly explains (A).
Consider a one-dimensional head on collision of two balls.
Assertion (A): The loss in kinetic energy of the system during the collision does not depend on the velocity of the observer.
Reason (R): Kinetic energy of a body is independent of velocity of observer.
Assertion (A): The loss in kinetic energy of a system is generally dependent on the observer's frame of reference. Thus, (A) is false.
Reason (R): Kinetic energy \(K = \frac{1}{2}mv^2\) depends on the velocity (v\), which is relative to the observer. Therefore, kinetic energy is dependent on the velocity of the observer. Thus, (R) is false.
Since both (A) and (R) are false, option (4) is correct.
Assertion (A): When one object collides with another object, the impulse during deformation and reformation will be in same direction on one particular object.
Reason (R): Due to deformation impulse the objects first deform and due to the same reformation impulse, they again try to regain its original shape.
Assertion (A): During collision, the impulses during the deformation phase and reformation phase on a particular object act in opposite directions. So, (A) is false.
Reason (R): The deformation impulse and reformation impulse are distinct. They are not the 'same' impulse. So, (R) is false. Since both (A) and (R) are false, option (4) is correct.
Assertion (A): Maximum energy loss occurs when the particles get stuck together as a result of collision.
Reason (R): A point particle of mass (m\) moving with speed (v\) collides with stationary point particle of mass (M\). Then the maximum energy loss possible is given \( \frac{m}{(m+M)}\left(\frac{1}{2}mv^2\right)\).
Assertion (A): Maximum kinetic energy loss occurs in a perfectly inelastic collision where particles stick together. So, (A) is true.
Reason (R): For a perfectly inelastic collision between mass (m\) (velocity (v\)) and stationary mass (M\), the energy loss is ( \Delta K = \frac{M}{(m+M)}\left(\frac{1}{2}mv^2\right)\). The given formula in (R) is incorrect.
So, (R) is false. Therefore, (A) is true and (R) is false. Option (3) is correct.