When a car moving at $30 \text{ m/s}$ hits a concrete wall, its momentum must be reduced to zero. The required impulse ($J = \Delta p$) is fixed by the car's mass and initial velocity. However, the force exerted on the passengers depends on the time duration of the impact ($F_{avg} = J / \Delta t$). Modern cars are designed with "crumple zones"—structural areas that deform predictably upon impact. This deformation significantly increases the collision time ($\Delta t$). By increasing the time it takes to stop, the average force exerted on the vehicle's occupants is drastically reduced, preventing fatal injuries.

In foundation engineering, large steel or concrete piles are driven into the ground using a heavy dropping weight called a hammer. The collision between the hammer and the pile is highly inelastic (often modeled with a low coefficient of restitution). While kinetic energy is lost to sound, heat, and deforming the top of the pile, momentum is conserved during the brief impact. Engineers use the principles of momentum conservation to calculate the velocity of the pile immediately after impact, which then allows them to determine how far the pile will penetrate the resistive soil based on work-energy principles.

A $0.145 \text{ kg}$ baseball is thrown horizontally towards home plate at $40 \text{ m/s}$. The batter hits it, and the ball leaves the bat horizontally in the opposite direction at $50 \text{ m/s}$. What is the magnitude of the impulse delivered to the ball by the bat?

If the bat in the previous example was in contact with the baseball for exactly $0.0015 \text{ seconds}$ (1.5 ms), what was the average force exerted by the bat on the ball?

A stationary $2 \text{ kg}$ block is struck by a mechanical hammer. The force exerted by the hammer over time is recorded and forms a triangular graph starting at $t=0$, peaking at $F_{max} = 500 \text{ N}$ at $t = 0.02 \text{ s}$, and returning to zero at $t = 0.05 \text{ s}$. What is the block's speed after the impact?

A $1500 \text{ kg}$ car traveling at $20 \text{ m/s}$ rear-ends a stationary $1000 \text{ kg}$ car at a stoplight. Their bumpers lock together. What is the velocity of the entangled cars immediately after the collision?

Ball A ($2 \text{ kg}$) moving at $5 \text{ m/s}$ to the right collides with Ball B ($3 \text{ kg}$) moving at $2 \text{ m/s}$ to the left. The coefficient of restitution for the collision is $e = 0.8$. Find the final velocities of both balls.

A $1200 \text{ kg}$ car traveling East at $15 \text{ m/s}$ collides at an intersection with a $1500 \text{ kg}$ truck traveling North at $10 \text{ m/s}$. The vehicles lock together upon impact. What is the magnitude and direction of their velocity immediately after the collision?