Suppose that a 40-kg runner in an imaginary world in which Planck's constant has the value 100 J · s must be located at the finish line to an accuracy of ± 0.33 m to be sure that it won the race.What would be the minimal uncertainty in its speed in that world?
A)0.12 m/s
B)0.13 m/s
C)0.60 m/s
D)1.2 m/s
E)15 m/s

Question

Quizplus · Accepted Answer

We can use the uncertainty principle to determine the minimal uncertainty in the runner's speed:
ΔxΔp ≥ ħ/2
where Δx is the uncertainty in the runner's position (±0.33 m), Δp is the minimal uncertainty in the runner's momentum (what we're solving for), and ħ is Planck's constant in this imaginary world (100 J · s).

The runner's momentum can be calculated as p = mv, where m is the mass and v is the velocity. We can assume that the runner is moving at a constant velocity, so the uncertainty in its momentum is Δp = mΔv.

Plugging in the given values, we get:

(0.33 m) m Δv ≥ (100 J · s) / 2

Solving for Δv, we get:

Δv ≥ (100 J · s)/(2(0.33 m)(40 kg))

Δv ≥ 0.60 m/s

Therefore, the minimal uncertainty in the runner's speed is 0.60 m/s, which corresponds to choice C.

Suppose That a 40-Kg Runner in an Imaginary World in Which