A bullet of mass m is fired into a block of mass Minitially...

A bullet of mass $m$ is fired into a block of mass Minitially at rest at the edge of a frictionless table of height $h$ Fig. P8.52 . The bullet remains in the block, and after impact the block lands a distance $d$ from the bottom of the table. Determine the initial speed of the bullet.

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Key Concepts

Projectile Motion

Projectile motion involves the motion of an object that is launched horizontally (or at an angle) and then moves under the influence of gravity. In this problem, after the collision, the block (with the bullet embedded) leaves the table with a horizontal velocity and undergoes free fall. The time of fall determined solely by the height of the table allows us to relate the horizontal distance traveled to the velocity of the block immediately after the collision.

Inelastic Collision

An inelastic collision is one in which the colliding objects stick together after impact, resulting in a loss of kinetic energy even though momentum is conserved. Here, the bullet embeds in the block, forming a single system whose combined mass moves with a reduced speed compared to the bullet’s initial speed. This concept is crucial for understanding how the initial velocities relate to the final velocity of the system.

Conservation of Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant. In the context of the problem, the horizontal component of momentum before and after the collision between the bullet and the block is conserved. This allows us to relate the bullet's initial momentum to the combined momentum of the bullet and block immediately after the collision.

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