A bullet of mass m is fired into a block of mass M initially...

A bullet of mass $m$ is fired into a block of mass $M$ initially at rest at the edge of a frictionless table of height $h$ (Fig. P9.57). 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.

Key Concepts

Conservation of Momentum

This concept states that in the absence of external forces, the total momentum of a system remains constant. In the context of the problem, it is applied during the collision between the bullet and the block. Since the bullet embeds into the block (an inelastic collision), momentum is conserved even though kinetic energy is not, allowing us to relate the bullet’s initial momentum to the combined momentum of the bullet and block immediately after the collision.

Inelastic Collision

An inelastic collision is one in which the colliding objects stick together after the impact. In such collisions, while momentum is conserved, kinetic energy is not. This concept is crucial because the bullet and block become a single object post-collision, requiring the use of momentum conservation to determine their common velocity immediately after impact.

Projectile Motion

After the collision, the block (with the embedded bullet) moves off the table and follows a projectile trajectory. Projectile motion analysis uses the kinematic equations, where the horizontal motion occurs at constant velocity and the vertical motion is influenced by gravity. The time to fall a given height, determined by gravitational acceleration, and the horizontal distance traveled are used together to back-calculate the initial velocity of the combined mass, and hence infer the bullet's initial speed.

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