The three air-track carts shown in Figure 7.19 have masses,...

The three air-track carts shown in Figure $7.19$ have masses, 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 $v$. All three carts are equipped with putty bumpers that make them have 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.

Key Concepts

Energy Loss in Inelastic Collisions

In completely inelastic collisions, some amount of the system's kinetic energy is converted into other forms of energy, leading to a lower total kinetic energy after the collision. Calculating the ratio of final to initial kinetic energy quantifies the efficiency of energy retention during the collision.

Conservation of Momentum

In any closed system where no external forces act, the total momentum remains constant regardless of the type of collision. This principle is crucial for analyzing collision problems where objects stick together, as it allows the determination of the final velocity of the combined mass by equating the total momentum before and after the collision.

Inelastic Collision

An inelastic collision is one in which the colliding objects do not bounce completely apart; in completely inelastic collisions, they stick together after impact. While momentum is conserved in these collisions, kinetic energy is not fully conserved due to transformation into other forms of energy, such as heat or deformation work.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the expression (1/2)mv^2. In collision problems, comparing initial and final kinetic energies can provide insight into energy losses resulting from non-conservative forces during impact.

System of Colliding Bodies

When analyzing multiple colliding bodies, especially when they stick together, the system can be treated as a single combined mass. This approach simplifies the problem by applying the conservation laws, such as momentum conservation, to the system as a whole rather than dealing with each body separately.

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