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.
Assertion (A): In case of bullet fired from a gun, the ratio of kinetic energy of gun and bullet is equal to ratio of masses of bullet and gun.
Reason (R): In firing of bullet, linear momentum of system is conserved.
Reason (R): For the bullet-gun system, the forces causing the bullet to fire are internal. Thus, linear momentum of the system is conserved. So, (R) is true.
Assertion (A): Let (m\) and (M\) be masses of bullet and gun, (v\) and (V\) their velocities. By momentum conservation, (mv = MV\). The ratio of kinetic energies is \( \frac{K_g}{K_b} = \frac{\frac{1}{2}MV^2}{\frac{1}{2}mv^2} = \frac{M(mv/M)^2}{mv^2} = \frac{m}{M}\). So, (A) is true.
(R) correctly explains (A) as the kinetic energy ratio is derived directly from momentum conservation. Option (1) is correct.