Answer
See explanation.
Work Step by Step
a. We know that the marble is somewhere on the table, so the maximum uncertainty in its position is 1.75 m.
b. The Heisenberg uncertainty principle limits our knowledge of its momentum.
$$\Delta x \Delta p_x \geq \frac{\hbar}{2}$$
$$\Delta x\;m \Delta v_x \geq \frac{\hbar}{2}$$
Find the minimum uncertainty.
$$ \Delta v_x =\frac{\hbar}{2m \Delta x}=3.01\times10^{-30}m/s$$
c. The uncertainty principle tells us that we cannot know for sure that the marble’s horizontal velocity is exactly zero. It could be as large as $3.01\times10^{-33}m/s$.
The longest time it could remain on the table is the time to travel the full 1.75 m, which is $t=\frac{1.75 m}{3.01\times10^{-33}m/s}=5.81\times10^{32}s$.
This is about $1.3\times10^{15}$ times older than the universe.
