A proton is accelerated to one- tenth of the velocity oflight. If its velocity can be measured with a precision + 1 %. What must be its uncertainity in position

To determine the uncertainty in the position of a proton accelerated to one-tenth the velocity of light, we can apply Heisenberg's Uncertainty Principle. This principle states that the product of the uncertainties in position and momentum of a particle cannot be smaller than a certain value, which is represented mathematically as Δx * Δp ≥ ħ/2, where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ (h-bar) is the reduced Planck's constant, approximately 1.055 x 10^-34 Js.

Understanding the Variables

First, let's break down the components we need:

• Velocity of the proton (v): Given as 1/10 the speed of light (c), where c ≈ 3 x 10^8 m/s. Therefore, v = 0.1c = 3 x 10^7 m/s.
• Mass of the proton (m): Approximately 1.67 x 10^-27 kg.
• Precision in velocity measurement (Δv): Given as ±1%, which translates to Δv = 0.01 * v = 0.01 * 3 x 10^7 m/s = 3 x 10^5 m/s.

Calculating the Uncertainty in Momentum

Momentum (p) is defined as the product of mass and velocity:

p = mv

Thus, the uncertainty in momentum (Δp) can be calculated using the formula:

Δp = m * Δv

Substituting the values we have:

Δp = (1.67 x 10^-27 kg) * (3 x 10^5 m/s) = 5.01 x 10^-22 kg·m/s.

Applying Heisenberg's Uncertainty Principle

Now, we can use the uncertainty principle to find the uncertainty in position (Δx):

Δx * Δp ≥ ħ/2

Rearranging gives us:

Δx ≥ ħ/(2Δp)

Substituting the values:

Δx ≥ (1.055 x 10^-34 Js) / (2 * 5.01 x 10^-22 kg·m/s)

Calculating this gives:

Δx ≥ (1.055 x 10^-34) / (1.002 x 10^-21) ≈ 1.05 x 10^-13 m.

Final Thoughts

Thus, the uncertainty in the position of the proton, when it is moving at one-tenth the speed of light with a velocity measurement precision of ±1%, is approximately 1.05 x 10^-13 meters, or 105 picometers. This illustrates how quantum mechanics imposes fundamental limits on our ability to precisely measure both the position and momentum of particles at high velocities.