You have entered a contest in which the contestants drop a...

You have entered a contest in which the contestants drop a marble with mass 20.0 g from the roof of a building onto a small target 25.0 $\mathrm{m}$ below. From uncertainty considerations, what is the typical distance by which you will miss the target, given that you aim with the highest possible precision? (Hint: The uncertainty $\Delta x_{f}$ in the $x$ -coordinate of the marble when it reaches the ground comes in part from the uncertainty $\Delta x_{i}$ in the $x$ -coordinate initially and in part from the initial uncertainty in $v_{x}$ . The latter gives rise to an uncertainty $\Delta v_{x}$ in the horizontal motion of the marble as it falls. The values of $\Delta x_{i}$ and $\Delta v_{x}$ are related by the uncertainty principle. A small $\Delta x_{i}$ gives rise to a large $\Delta v_{x}$ and vice versa. Find the value of $\Delta x_{i}$ that gives the smallest total uncertainty in $x$ at the ground. Ignore any effects of air resistance.)

You have entered a contest in which the contestants drop a marble with mass 20.0 g from the roof of a building onto a small target 25.0 $\mathrm{m}$ below. From uncertainty considerations, what is the
typical distance by which you will miss the target, given that you aim with the highest possible precision? (Hint: The uncertainty $\Delta x_{f}$ in the $x$ -coordinate of the marble when it reaches the ground
comes in part from the uncertainty $\Delta x_{i}$ in the $x$ -coordinate initially and in part from the initial uncertainty in $v_{x}$ . The latter gives rise to an uncertainty $\Delta v_{x}$ in the horizontal motion of the marble as it falls. The values of $\Delta x_{i}$ and $\Delta v_{x}$ are related by the uncertainty principle. A small $\Delta x_{i}$ gives rise to a large $\Delta v_{x}$ and vice versa. Find the value of $\Delta x_{i}$ that gives the smallest total uncertainty in $x$ at the ground. Ignore any effects of air resistance.)

Key Concepts

Error Propagation in Projectile Motion

In projectile motion, initial uncertainties in quantities such as the position and velocity of a projectile can propagate and affect the final outcome, such as its landing position. Here, small uncertainties in the initial horizontal position and initial horizontal velocity grow over time, depending on the duration of the fall. Understanding how these uncertainties combine is essential for predicting where a projectile might land under realistic measurement limitations.

Heisenberg Uncertainty Principle

This principle in quantum mechanics sets a fundamental limit on the simultaneous precision with which certain pairs of physical properties, such as position and momentum, can be known. It states that the product of the uncertainties in position and momentum must be at least on the order of ?/2. In contexts where precision is important, this principle establishes an inherent trade-off: attempting to measure one variable with greater precision necessarily increases the uncertainty in its conjugate partner.

Optimization of Uncertainty Trade-Off

When two sources of error contribute to a final measurement, as in the case of initial position and velocity uncertainties, there is an optimal balance where the combined uncertainty is minimized. This involves adjusting the precision of one measurement relative to the other to achieve the lowest possible total uncertainty. The optimization process in this context showcases how fundamental limits (like those set by the uncertainty principle) impact practical measurements and outcomes.

Transcript

00:01 In this exercise, we have a marble that has a mass of 20 grams and is launched from a height of 25 meters, and after that it falls freely to the ground.

00:13 And we want to know by which distance you miss the target, if you're the person who's throwing the marbles in the target on the ground, if you're aiming for the highest precision that you can possibly achieve.

00:31 Okay, so first, the first information that we are going to need in this, in order to solve this question, is the time it takes for the marble to fall to the ground.

00:44 And remember that according to newtonian mechanics, the time is just the square root of two times g times the height.

00:58 So this is the time.

00:59 I'm not going to substitute the numbers just yet.

01:02 So let's calculate the uncertainty in the position, the final uncertainty in the position given the initial uncertainty in the position.

01:17 So we know that we have that the final uncertainty in the position of the marble is equal to the initial uncertainty in the position, plus the uncertainty in the x component of the speed times t.

01:32 This is just the equation for a uniform motion in the x direction.

01:41 Notice that there are no forces in the x direction.

01:43 And you're aiming for the highest precision.

01:46 So you want delta xf, the uncertainty in the x direction, to be in minimum.

01:54 So if you want this to be minimum, then we have to differentiate it.

02:03 And i'm going to differentiate it with respect to delta xi.

02:06 And we want this derivative to be zero, okay? then we're finding a minimum of the uncertainty...