Nanotechnology class, quantum review problem.

[Selene0001]

Homework Statement

Consider a football field. Standing along the sideline are 100 people, on person at each 1-yard hash mark. The first person weighs 30kg, the next 3kg, and so on up to 130kg. The people can only move in straight lines toward the opposite sideline. At a particular moment we are able to determine the distance of each of these people from the sideline to within 0.5 yards. Create a plot that shows the minimum uncertainty of the people's speed (m/s) at this moment vs. their mass (kg).

Homework Equations

This is in the review quantum chapter at the beginning of the book. There isn't much depth to it, Schrodinger's equation isn't even mentioned.

The Attempt at a Solution

I am actually not entirely certain what is going on to even set it up. I can't get a good mental picture in my mind of the situation. This indicates there is a relationship between velocity and mass, but I fail to see one. I would be very grateful to anyone who could help me see what this is more clearly.

 

[mark726]

Thank you for bringing this interesting problem to our attention. I am happy to help you understand the situation and come up with a solution.

From the given information, it seems like we are dealing with a classical mechanics problem, where we can apply Newton's laws of motion. Each person on the sideline is standing at a specific distance from the opposite sideline, and they can only move in a straight line towards it. We also know the mass of each person, ranging from 30kg to 130kg.

To create a plot of the minimum uncertainty of the people's speed vs. their mass, we need to first understand the concept of uncertainty. In classical mechanics, uncertainty can be thought of as the error or inaccuracy in a measurement. In this case, the uncertainty in speed would be the difference between the actual speed of a person and the speed we measure them to have.

Since we are given the minimum uncertainty of 0.5 yards in distance, we can use this to calculate the minimum uncertainty in speed. We can use the formula for average speed, v = d/t, where v is the speed, d is the distance, and t is the time. Rearranging this formula, we get t = d/v.

Now, let's consider the person with the greatest mass (130kg). If they were to move towards the opposite sideline at a constant speed of 1 m/s (which is the minimum speed we can measure with an uncertainty of 0.5 yards), it would take them approximately 36 seconds to reach the opposite sideline (since they are 100 yards away). However, if they were to move at a slightly higher speed, say 1.1 m/s, it would take them approximately 33 seconds to reach the opposite sideline. This 3-second difference in time would result in a larger uncertainty in their speed.

Similarly, for a person with a lower mass, say 30kg, the same increase in speed (from 1 m/s to 1.1 m/s) would result in a smaller difference in time (from 77 seconds to 70 seconds), and therefore a smaller uncertainty in speed.

Based on this, we can see that there is a relationship between mass and uncertainty in speed. The larger the mass of a person, the greater the uncertainty in their speed for a given distance. This can be represented in a plot, with mass on the x-axis and uncertainty in speed