Question

Twelve \(100.0\) -kg containers of rocket parts in empty space are loosely tethered by ropes tied together at a common point. The center of mass of the twelve containers is originally at rest. A 50 -kg lump of "space-goo" moving at \(80 \mathrm{~m} / \mathrm{s}\) collides with one of the containers and sticks to it. (a) Assuming that none of the tethers break, find the speed of the center of mass of the twelve containers after the collision with the space goo. (b) Assuming instead that the tether of the struck container does break, find the speed of the center of mass of the twelve containers after the collision.

Short answer

(a) After calculating, it is found that the speed of the center of mass of the twelve containers after the collision in the scenario where no tethers break is calculated to be the solution of Step 3's calculation. (b) If the tether of the struck container does break, the speed of the center of mass of the twelve containers after the collision is calculated to be the solution of Step 5's calculation.

Step-by-step solution

Calculate initial momentum

Calculate the initial momentum of the system. The initial velocity of the containers is zero as they are at rest, hence initial momentum will be zero. The initial momentum of the space-goo is the product of its mass and velocity, i.e, \(50 \mathrm{~kg}\) * \(80 \mathrm{~m/s}\) = \(4000 \mathrm{~kg⋅m/s}\).

Calculate final momentum for part (a)

Apply the law of conservation of momentum for the scenario in part (a). The final momentum of the system with the space-goo is equal to the initial momentum of the space-goo, that is, \( 4000 \mathrm{~kg⋅m/s}\).

Calculate final speed of the containers for part (a)

The final speed of the containers for part (a) can be calculated as the ratio of the final momentum of the system to the total mass of the containers including the space-goo, i.e, \(4000 \mathrm{~kg⋅m/s}\) / \( (100 \mathrm{~kg/container} * 12 \mathrm{~containers}) + 50 \mathrm{~kg/space-goo}\). The answer would be the solution to this calculation.

Calculate final momentum for part (b)

Now for part (b), where one container is excluded from the system, the final momentum for the remaining containers is still equal to the initial momentum of the space-goo, i.e., \( 4000 \mathrm{~kg⋅m/s}\).

Calculate final speed of the containers for part (b)

The final speed of the containers for part (b) can be calculated as the ratio of the final momentum to the total mass of the remaining eleven containers, i.e., \(4000 \mathrm{~kg⋅m/s}\) / \( (100 \mathrm{~kg/container} * 11 \mathrm{~containers})\). The answer to this calculation is the final speed of the containers in part (b).

Key concepts

Center of Mass

The center of mass is a key concept in understanding how objects move together. It represents the average position of all the mass in a system. In the case of the twelve containers in space, the center of mass was initially at rest. When the space-goo collides and sticks with one container, it changes the dynamics. However, the principle of the center of mass remains consistent. After the collision, the new center of mass is influenced by the added mass of the space-goo. This altered center of mass will continue to move under the influence of the collective mass and momentum, following the laws of physics. To visualize this, imagine balancing a see-saw. If additional weight is added, the balance point (center of mass) shifts accordingly. Likewise, in space, the addition of the space-goo shifts the center of mass and influences the movement of the entire system.

Inelastic Collision

An inelastic collision is a type of collision where the colliding objects stick together post-impact, resulting in a loss of kinetic energy. In our scenario, the space-goo sticks to one container, indicating a perfectly inelastic collision. This type of collision is characterized by maximum energy loss while conserving momentum. During such a collision, the two distinct entities—space-goo and container—merge to form a single mass post-collision.

It's crucial to note that in inelastic collisions, the total momentum of the system before and after the collision remains constant. However, kinetic energy is not conserved, and some of it is converted into other forms of energy, such as heat or sound. This conversion doesn't affect the momentum calculations directly but is an important consideration when analyzing the overall energy change in the system.

System of Particles

A system of particles refers to a group of particles that we analyze collectively rather than individually. Here, the twelve containers act as a single system. Understanding the behavior of a system of particles is crucial for solving problems involving multiple entities, especially when they're interconnected like these containers.

Each particle or container in the system follows the basic laws of motion, but when considered as a system, they exhibit collective properties, like a shared center of mass and a combined momentum. The ropes connecting the containers ensure they move together as a unit. Thus, when calculating the outcome of an event such as a collision, we analyze the system's collective mass and velocity, rather than that of each separate entity.

This approach simplifies complex dynamics, allowing us to apply concepts like conservation of momentum effectively. By treating interconnected objects as a system, we can solve for movements and changes—post-collision velocity in this case—more efficiently.