A block of mass m is pushed with a veloeity u towards a...

A block of mass $m$ is pushed with a veloeity $u$ towards a movable wedge of mass $n m$ and height $h$, figure. All the surfaces are smooth. The minimum value of $u$ for which the block will reach the top of wedge is (a) $\sqrt{2 g h\left(1-\frac{1}{n}\right)}$ (b) $\sqrt{2 g h\left(1+\frac{1}{n}\right)}$ (c) $\sqrt{3 g h}$ (d) $\sqrt{2 g h}$

A block of mass $m$ is pushed with a veloeity $u$ towards a movable wedge of mass $n m$ and height $h$, figure. All the surfaces are smooth. The minimum value of $u$ for which the block will reach the top of wedge is
(a) $\sqrt{2 g h\left(1-\frac{1}{n}\right)}$
(b) $\sqrt{2 g h\left(1+\frac{1}{n}\right)}$
(c) $\sqrt{3 g h}$
(d) $\sqrt{2 g h}$

Key Concepts

Kinetic and Potential Energy

Kinetic energy (associated with the motion of the block and wedge) and potential energy (associated with the height gained by the block on the wedge) are critical in setting up the energy balance equations. Understanding how these forms of energy convert into each other is key to determining the minimum initial speed required for the block to reach the top of the wedge.

Relative Motion

Relative motion concepts help in understanding and analyzing the movement of one object with respect to another, rather than with respect to the ground. In the block-and-wedge scenario, it is useful to examine how the block moves relative to the wedge, considering that the wedge itself can move due to the absence of friction.

Conservation of Momentum

In systems where no external forces act in a particular direction, the momentum in that direction is conserved. For a frictionless block-and-wedge system, horizontal momentum is conserved, meaning that the initial momentum of the block influences the motion of both block and wedge as they interact.

Conservation of Energy

This principle states that the total mechanical energy of a closed system remains constant if there are no non-conservative forces (like friction) doing work. In problems involving a block and a wedge with smooth surfaces, both kinetic and potential energies are considered, and their transformation determines the conditions required for the block’s motion up the wedge.

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