A proton having mass equal to 1.66 10⁻²⁷ ~kg is accelerated to one tenth of the velocity of light. If its velocity can be measured to an accuracy of 2 % , what would be the uncertainty in its position?

Step-by-step solution

Given mass of proton m = 1.66 10⁻²⁷ kg . Velocity v = 1 10 c = 1 10 3 10⁸ m/s = 3 10⁷ m/s . Uncertainty in velocity v = 2 % of v = 0.02 3 10⁷ = 6 10⁵ m/s . According to Heisenberg's uncertainty principle, x p h 4 , where p = m v . Therefore, x h 4 m v . Substituting the values h = 6.626 10⁻³⁴ J s , m = 1.66 10⁻²⁷ kg , and v = 6 10⁵ m/s : x 6.626 10⁻³⁴ 4 3.1416 1.66 10⁻²⁷ 6 10⁵ . x 6.626 10⁻³⁴ 12.5664 1.66 6 10⁻²² . x 6.626 10⁻³⁴ 125.162 10⁻²² 0.05294 10⁻¹² m = 5.294 10⁻¹⁴ m . Answer: 5.29 10⁻¹⁴ ~m