A pesky $1.5$-mg mosquito is annoying you as you attempt to study physics in your room, which is $5.0$ m wide and $2.5$ m high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit.
(a) What is the maximum uncertainty in the horizontal position of the mosquito?
(b) What limit does the Heisenberg uncertainty principle place on your ability to know the horizontal velocity of this mosquito? Is this limitation a serious impediment to your attempt to swat it?

Verified step by step guidance

1. Step 1: Understand the problem and identify the key concepts. This problem involves the Heisenberg Uncertainty Principle, which states that the uncertainty in position (Δx) and the uncertainty in momentum (Δp) are related by the inequality Δx * Δp ≥ ℏ / 2, where ℏ is the reduced Planck's constant (ℏ ≈ 1.055 × 10⁻³⁴ J·s). Momentum is related to velocity by p = m * v, where m is the mass and v is the velocity.
2. Step 2: Calculate the maximum uncertainty in the horizontal position of the mosquito. Since the mosquito is flying in a room that is 5.0 m wide, the maximum uncertainty in its horizontal position (Δx) can be taken as the width of the room, Δx = 5.0 m.
3. Step 3: Use the Heisenberg Uncertainty Principle to find the minimum uncertainty in the horizontal momentum (Δp). Rearrange the inequality to solve for Δp: Δp ≥ ℏ / (2 * Δx). Substitute the known values of ℏ and Δx into this equation to calculate Δp.
4. Step 4: Relate the uncertainty in momentum (Δp) to the uncertainty in velocity (Δv). Since momentum is given by p = m * v, the uncertainty in velocity can be expressed as Δv = Δp / m. Use the mosquito's mass (m = 1.5 mg = 1.5 × 10⁻⁶ kg) and the previously calculated Δp to find Δv.
5. Step 5: Analyze whether the uncertainty in velocity (Δv) is significant. Compare the calculated Δv to the typical speed of a mosquito (which is on the order of 1 m/s). If Δv is much smaller than the mosquito's speed, the uncertainty is negligible, and the Heisenberg Uncertainty Principle does not pose a serious impediment to swatting the mosquito.

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and exact momentum (or velocity) of a particle. This principle highlights a fundamental limit in quantum mechanics, where the more precisely one property is known, the less precisely the other can be known. For small particles like a mosquito, this principle becomes significant when estimating their position and velocity.

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. In physics, momentum is crucial because it is conserved in isolated systems, meaning the total momentum before and after an event remains constant. Understanding momentum helps in analyzing the motion of objects, including the mosquito in this scenario, and is essential for calculating its velocity.

Measurement Uncertainty

Measurement uncertainty refers to the doubt that exists about the result of any measurement. In the context of the mosquito's position and velocity, it quantifies the limits of precision in determining these values. This uncertainty can arise from various factors, including the limitations of measuring instruments and the inherent variability in the system being measured, which is particularly relevant when applying the Heisenberg Uncertainty Principle.