In the figure, the block B of mass m starts from rest at the...

In the figure, the block $B$ of mass $m$ starts from rest at the top of a wedge $W$ of mass $M .$ All surfaces are smooth, $W$ can slide on the ground, $B$ slides down onto the ground, moves along it with a speed $y$, has an elastic collision with the wall, and climbs back on to $W$ (a) From the beginning, till the collision with the wall, the centre of mass of ' $B$ plus $W$ ' does not move horizontally. (b) After the collision, the centre of mass of $" B$ plus $W^{\prime}$ moves with the velocity $\frac{2 m}{\operatorname{m} \mid M}$ (c) When $B$ reaches its highest position of $W$, the speed of $W$ is $\frac{2 m v}{m \wedge M}$ (d) All of these

In the figure, the block $B$ of mass $m$ starts from rest at the top of a wedge $W$ of mass $M .$ All surfaces are smooth, $W$ can slide on the ground, $B$ slides down onto the ground, moves along it with a speed $y$, has an elastic collision with the wall, and climbs back on to $W$
(a) From the beginning, till the collision with the wall, the centre of mass of ' $B$ plus $W$ ' does not move horizontally.
(b) After the collision, the centre of mass of $" B$ plus $W^{\prime}$ moves with the velocity $\frac{2 m}{\operatorname{m} \mid M}$
(c) When $B$ reaches its highest position of $W$, the speed of $W$ is $\frac{2 m v}{m \wedge M}$
(d) All of these

Key Concepts

Center of Mass

The center of mass of a system is the point at which the entire mass of the system can be considered to be concentrated. In mechanics, especially when external forces like friction or applied forces are absent or cancel out, the center of mass follows a predictable path based solely on the net external forces. This concept is critical when analyzing how parts of a system move relative to each other, even if there is internal motion or collisions.

Conservation of Momentum

The law of conservation of momentum states that in a closed system with no net external forces, the total momentum—both magnitude and direction—remains constant. This principle applies to systems where internal forces, like those during collisions or along inclined planes, redistribute momentum among components while the overall momentum stays unchanged. It is particularly important in analyzing collision scenarios and the overall motion of combined systems.

Elastic Collision

An elastic collision is a type of collision in which both momentum and kinetic energy are conserved. In such collisions, no energy is lost to sound, heat, or deformation, allowing precise prediction of the velocities post-collision using conservation laws. This concept is essential to understand the dynamic behavior of objects after impact, as it affects subsequent motions and interactions with other system components.

Dynamics of Systems with Variable Mass Distribution

In problems involving moving objects on a wedge or similar setups, the changing distribution of mass is key to understanding the system’s dynamics. Such problems require careful application of conservation principles, as different parts of the system may move relative to each other without external horizontal forces affecting the overall center of mass. This analysis helps explain seemingly counterintuitive motions such as wedges moving in reaction to sliding objects.

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