A mosquito of mass $\mathbf{\text{0.15mg}}$is found to be flying at a speed of $\mathbf{\text{50cm/s}}$within an uncertainty of $\mathbf{\text{0.5mm/s}}$.(a) How precisely may its position be known? (b) Does this inherent uncertainty present any hindrance to the application of classical mechanics?

A mosquito of mass $0.15mg$

Speed of $0.15mg$

Uncertainty of$0.5mm/s$

Given the uncertainty in the velocity, we need to work out the limit on the position measurement ($\text{Δx}$).

$\because \text{ΔpΔx}\ge \frac{\hslash }{\text{2}}\Rightarrow \text{mΔvΔx}\ge \frac{\hslash }{\text{2}}$

$\therefore \text{mΔvΔx}\ge \frac{\hslash }{\text{2}}$

$\therefore {\text{0.15×10}}^{\text{-6}}{\text{kg×0.50×10}}^{\text{-3}}\text{m/s×Δx}\ge \frac{{\text{1.054×10}}^{\text{-34}}\frac{{\text{kg×m}}^{\text{2}}}{\text{s}}}{\text{2}}$

.$\therefore \text{Δx}\ge \frac{{\text{1.054×10}}^{\text{-34}}\text{m}}{{\text{2×7.5×10}}^{\text{-11}}}$

Mosquitoes have a very low positional uncertainty, which is a feature of particles in general. In other words, the mosquito's associated wave is so little that the classical treatment of its motion provides a very close approximation to its true motion. To begin addressing the mosquito's wave nature, its mass must be at least somewhat comparable to that of the electron. As a result, the conclusion is that the uncertainty principle does not obstruct the classical treatment in this circumstance.