Question

Two skaters, one with mass \(65 \mathrm{~kg}\) and the other with mass \(42 \mathrm{~kg}\), stand on an ice rink holding a pole with a length of \(9.7 \mathrm{~m}\) and a mass that is negligible. Starting from the ends of the pole, the skaters pull themselves along the pole until they meet. How far will the 42 -kg skater move?

Short answer

The 42-kg skater will move approximately 5.89 meters.

Step-by-step solution

Understanding the Law of Conservation of Momentum

The law of conservation of momentum states that the total momentum before and after an event must be the same if no external forces are acting on the system. This law applies to this exercise because there are no external forces acting on the skaters and the pole, making it an isolated system. The skaters' movements will equalize their individual momentums, resulting in a joint center of mass that remains stationary. Since their momentums are equal but opposite in direction, they cancel each other out when added, which keeps the total momentum at zero, in line with the conservation law.

Formulas and Variables

The distances the skaters move are inversely proportional to their masses. This gives us the formula \(d_1 = \frac{m_2 \cdot d}{m_1+m_2}\) and \(d_2 = \frac{m_1 \cdot d}{m_1+m_2}\), where \(d_1\) is the distance moved by skater 1, \(m_2\) is the mass of skater 2, and \(d\) is the total length of the pole. Subsequently, \(d_2\) is the distance moved by skater 2, and \(m_1\) is the weight of skater 1. The total distance \(d\) is the length of the pole that they start from its ends and slide towards each other.

Calculate Distance

Since we want to find how far does the 42-kg skater (skater 2) move, we will use the formula for \(d_2 = \frac{m_1 \cdot d}{m_1+m_2}\). Substituting the given numbers in the formula, \(d_2 = \frac{65kg \cdot 9.7m}{65kg+42kg}\) which simplifies to \(d_2 = \frac{630.5}{107}\). Solve this equation to get the final result.

Key concepts

Physics Problem Solving

Physics problems, such as the scenario involving two skaters on an ice rink, require a logical approach rooted in understanding physical concepts and laws. First and foremost, the key to successful problem solving in physics is comprehending the principles involved. In this case, the law of conservation of momentum is paramount.
When tackling such a problem, it's critical to identify the system in question—here, it's the skaters and the pole. Recognizing that they form an isolated system with no external forces at play allows one to assert that momentum will be conserved. Next, translating the qualitative understanding of the problem into a quantitative one involves identifying the correct formulas, as well as the variables involved.
Finally, solving the problem turns into a mathematical exercise, where known values are substituted into the formulas, and unknowns are calculated. Ensuring that you clearly show each step of the calculation, as demonstrated in our example, not only helps avoid simple errors but also improves understanding of the relationship between the variables and the physics principle at play.

Momentum in Collisions

The momentum of an object is defined as the product of its mass and velocity. In collisions or interactions like the one between the two skaters, momentum plays a crucial role in determining the outcome. For systems where no external forces are involved, the law of conservation of momentum states that the total momentum remains constant.
In the context of the skaters, this concept means that the momentum one skater loses by moving towards the center is gained by the other skater. If we consider the skaters and the pole to be heading towards a collision of sorts-lacking any external interference-their combined momentum before and after they meet must remain the same.

Equality of Momentum

This balance or equality of momentum allows us to set up specific mathematical relationships that lead to the formulas given in the solution. Understanding how to manipulate these formulas is critical for solving any problem involving momentum in collisions.

Center of Mass

The center of mass is a point that represents the average position of all the mass in a system. For a system to remain stationary, any movement of parts of the system must occur in such a way that the center of mass does not change position. This happens if the system is isolated with no external forces, as is the case with our skaters.
While the skaters pull themselves towards each other, they are actually moving towards their shared center of mass. The movement is inversely proportional to their mass—the lighter skater will move more than the heavier one. By understanding the relationship between mass and distance traveled, we assert that the center of mass remains in the same location despite their individual movements.
Utilizing the concept of center of mass in combination with the law of conservation of momentum can greatly simplify the analysis of many physical problems. This approach transforms a potentially complex scenario into one where the distances each skater moves can be calculated with straightforward formulas and arithmetic.