To find the uncertainty in the position of the proton, we use the Heisenberg Uncertainty Principle, which in terms of position
(x) and momentum
(p) is expressed as:
∆x∆p≥‌Where
ħ (the reduced Planck constant) is approximately
1.05×10−34‌Js.
First, let's calculate the momentum of the proton when it's moving at one tenth of the velocity of light
(c=3.00×108m∕ s).
The velocity of the proton,
v=0.1c=0.1×3.00×108m} s=3.00×107m} s
The momentum
p of the proton is given by:
p=mvwhere
m=1.66×10−27‌kg (mass of the proton) and
v=3.00×107m} s.
So,
p=(1.66×10−27‌kg)×(3.00×107m} s)=4.98×10−20‌kg⋅m} s
Next, we find the uncertainty in momentum
(∆p) using the given accuracy in measuring velocity:
Given that velocity can be measured to an accuracy of
±2% :
∆v=0.02×3.00×107m∕ s=6.00×105m} s
So,
∆p=m∆v=(1.66×10−27‌kg)×(6.00×105m} s)=9.96×10−22k
We now substitute
∆p into the Heisenberg Uncertainty Principle:
∆x∆p≥‌ ‌∆x≥‌‌∆x≥‌| 1.05×10−34‌Js |
| 2×9.96×10−22‌kg⋅m} s |
‌∆x≥‌| 1.05×10−34 |
| 1.992×10−21 |
m‌∆x≥5.27×10−14mThis calculation gives the uncertainty in the proton's position as approximately
5.27×10−14m, which closely matches option B: