The three air carts shown in FIGURE 9-44 have masses,...

The three air carts shown in FIGURE $9-44$ have masses, reading from left to right, of $m, 2 m,$ and $4 m,$ respectively. Initially, the cart on the right is at rest, whereas the other two carts are moving to the right with a speed $v_{0}$ . All carts are equipped with putty bumpers that give completely inelastic collisions. (a) Find the final speed of the carts. (b) Calculate the ratio of the final kinetic energy of the system to the initial kinetic energy.

The three air carts shown in FIGURE $9-44$ have masses, reading from left to right, of $m, 2 m,$ and $4 m,$ respectively. Initially, the cart on the right is at rest, whereas the other two carts are moving to
the right with a speed $v_{0}$ . All carts are equipped with putty bumpers that give completely inelastic collisions. (a) Find the final speed of the carts. (b) Calculate the ratio of the final kinetic energy of the system to the initial kinetic energy.

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Key Concepts

Conservation of Momentum

In any isolated system, the total linear momentum remains constant during interactions such as collisions. This principle is used to determine the final velocity of colliding bodies by ensuring that the sum of the individual momenta before the collision equals the momentum of the combined mass after the collision.

Completely Inelastic Collisions

In a completely inelastic collision, the colliding objects stick together after impact and move with a common velocity. While momentum is conserved in these collisions, kinetic energy is not; some of it is transformed into other forms of energy, such as heat or deformation.

Kinetic Energy Considerations in Inelastic Collisions

Although momentum conservation provides a means to calculate the resulting speed after a collision, the kinetic energy is not conserved in completely inelastic collisions. The analysis of the ratio of final to initial kinetic energy highlights the energy lost during the collision process and helps in understanding the efficiency of momentum transfer.

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