You have entered a contest in which the contestants drop a marble with mass 27.0g from the roof of a building onto a small target 23.0m 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?
To calculate the typical distance by which you will miss the target, we need to consider the uncertainty in the initial position and the uncertainty in the final position.
1. Uncertainty in the initial position:
You mentioned that you aim with the highest possible precision, so let's assume that the uncertainty in the initial position is negligible. In other words, we assume that the marble is dropped exactly from the roof of the building.
2. Uncertainty in the final position:
The uncertainty in the final position is influenced by factors such as the air resistance, slight variations in the marble's mass, and the exact trajectory of the marble as it falls. This uncertainty is more challenging to calculate precisely.
However, we can estimate the typical distance by which you will miss the target by considering the vertical distance and the lateral distance traveled by the marble.
Vertical distance:
The vertical distance is 23.0m since the target is 23.0m below the roof.
Lateral distance:
To estimate the lateral distance, we can use the formula for the horizontal distance traveled by a projectile when there is no air resistance:
d = v * t,
where:
- d is the lateral distance,
- v is the horizontal velocity of the marble,
- t is the time of flight.
Horizontal velocity:
The marble only has a vertical velocity when dropped vertically, and its horizontal velocity is assumed to be zero. Hence, we can assume that the horizontal velocity (v) is negligible.
Time of flight:
To calculate the time of flight (t), we can use the equation of motion for vertical motion:
h = (1/2) * g * t^2,
where:
- h is the vertical distance,
- g is the acceleration due to gravity (approximately 9.8 m/s²),
- t is the time of flight.
Rearranging the equation, we have:
t = sqrt(2h / g).
Substituting the given values, we have:
t = sqrt(2 * 23.0 m / 9.8 m/s²) = 2.42 s (rounded to two decimal places).
Lateral distance:
Since the horizontal velocity is negligible, the lateral distance (d) will also be negligible.
Therefore, considering the given assumptions and calculations, the typical distance by which you will miss the target (in the lateral direction) would be negligible for the described scenario.
If you use the Heisenberg uncertainty principle, the minimum error will be something like
sqrt[(h*R/(2 pi))/p]
where p is the nominal momentum at impact and R is the distance it falls.
You must be careful to recognized that the lateral position error depends upon the lateral momentum uncertaintly. You need to relate both to the angular deviation from a vertical fall.
It is more likely that the minimum position error will be governed by random turbulent eddy production in the wake of the marble, which has nothing to do with quantum mechanics. This goes beyond the scope of your physics question.