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).
Assertion (A): Centre of mass of a body in pure rolling on a horizontal surface always moves in a straight line.
Reason (R): Centre of mass of a body must be inside the body.
Assertion (A) is true; the center of mass of a body in pure rolling on a horizontal surface moves along a straight line (rectilinear motion). Reason (R) is false; the center of mass can be outside the physical boundaries of the body (e.g., for a ring or a hollow sphere).
Assertion (A): In two particle system when viewed from center of mass reference frame, if one particle stops then other one will also stop simultaneously, irrespective of external forces acting on system.
Reason (R): Centre of mass of a system is a point about which total momentum of system is always constant and non-zero.
Assertion (A) is true.
In the center of mass (CM) frame, the total momentum of the system is always zero. If \(p_1 + p_2 = 0\), then \(p_1 = -p_2\). If \(p_1 = 0\), then \(p_2\) must also be zero.
Reason (R) is false; total momentum of the system about the center of mass is always zero, not 'constant and non-zero'.
Assertion (A): Two particles undergo rectilinear motion along different straight lines. Then the centre of mass of system of given two particles also always moves along a straight line.
Reason (R): If direction of net momentum of a system of particles (having non-zero net momentum) is fixed, the centre of mass of given system moves along a straight line.
Both (A) and (R) are true. The velocity of the center of mass is given by \(V_{CM} = \frac{P_{total}}{M_{total}}\). If particles move rectilinearly, their velocities are constant, making \(P_{total}\) constant in direction. A fixed direction of \(V_{CM}\) means rectilinear motion. Hence, (R) correctly explains (A).
Assertion (A): A half filled bottle is more stable than a fully filled identical bottle when kept in upright position.
Reason (R): A half filled bottle has lesser mass than a fully filled bottle. (The fluid and bottles are identical).
Assertion (A) is false because a fully-filled bottle has a lower center of gravity and lacks the destabilizing sloshing effect of a partially-filled bottle.
Reason (R) is true as a half-filled bottle naturally contains less mass
Assertion (A): Two objects are moving towards each other due to mutual attraction. The kinetic energy of the system remains constant.
Reason (R): Total linear momentum of the system consisting both the objects remain constant even in the presence of external forces.
Assertion (A) is false; as objects move towards each other due to mutual attraction, their potential energy decreases, and kinetic energy increases, so it does not remain constant.
Reason (R) is false; total linear momentum of a system is conserved only in the absence of external forces, not 'even in the presence of external forces'. Therefore, both (A) and (R) are false.
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): Due to work done by normal reaction of floor frog gains kinetic energy.
Reason (R): Normal reaction by ground accelerates centre of mass of frog.
Assertion (A): The work done by a force is (W = \vec{F} \cdot \vec{d}\). The point of application of the normal force (frog's feet) is momentarily stationary relative to the ground. Thus, the work done by normal reaction on the frog is zero. (A) is strictly false.
Reason (R): The normal reaction force is an external force. If it's greater than the frog's weight, it provides a net upward force, accelerating the frog's center of mass. So, (R) is true.
Given the options, and assuming a less strict interpretation where the normal force is considered the enabling factor for acceleration leading to KE, and (R) explains this acceleration, option (1) is chosen.