You're with 19 other people on a boat at rest in frictionless water. The group's total mass is \(1500 \mathrm{kg}\), and the boat's mass is \(12,000 \mathrm{kg} .\) The entire party walks the \(6.5-\mathrm{m}\) distance from bow to stern. How far does the boat move?

Momentum in physics is a measurement of the quantity of motion of a moving body and is calculated as the product of its mass and velocity. In the context of our boat example, the momentum of the group of people can be quantified by considering the mass of the group and their speed as they move from one end of the boat to the other. Since velocity includes direction, momentum is a vector quantity, meaning it has both magnitude and direction.

In simpler terms, momentum helps us understand how challenging it's going to be to stop that group of people once they start moving. The larger their combined mass or the faster they move, the more momentum they have. If they were to suddenly stop, the energy of their movement would have to go somewhere, which brings us to how their momentum affects the boat, a scenario perfectly captured by the law of conservation of momentum.

A boat at rest in frictionless water is an idealized scenario that helps us focus on the physics without the complicating factors of real-world friction. In this case, the water's lack of resistance means that any motion initiated by the boat's occupants will be more evident in the boat's reaction. For our exercise, when the group of 19 people starts moving, there's nothing to stop the boat from reacting because there's no friction to hold it in place.

It's analogous to the slickness of ice. Imagine trying to walk on a frozen pond; as you slide forward, the ice would seem to move backward under your feet. In the case of the boat, that back-and-forth dance between the people and the boat is due to the conservation of momentum, without any interference from water resistance.

The law of conservation of momentum states that if no external forces are acting on a system of objects, the total momentum of the system remains constant. Applying this law to our boat example, we see that the initial state of the system—a stationary boat with people onboard—has a total momentum of zero because none of the objects are moving.

As the group moves to the stern, their forward momentum is balanced by the boat's backward momentum, ensuring that the total momentum of the system remains unchanged—that is, still zero. This exchange of momentum is why the boat moves backward when the party moves forwards. The distances the group and the boat move are inversely related to their masses, and the conservation of momentum allows us to calculate precisely how far the boat travels in reaction to the group's movement.