An accelerated electron has a speed of $$5 \times 10^6$$ ms$$^{-1}$$ with an uncertainty of 0.02%. The uncertainty in finding its location while in motion is $$x \times 10^{-9}$$ m. The value of $$x$$ is _________. (Nearest integer)
[Use mass of electron = $$9.1 \times 10^{-31}$$ kg, h = $$6.63 \times 10^{-34}$$ Js, $$\pi$$ = 3.14]

We have an electron of mass $$m = 9.1 \times 10^{-31}\,{\rm kg}$$ moving with a speed $$v = 5 \times 10^{6}\,{\rm m\,s^{-1}}$$. The problem states that the speed is known only to within an uncertainty of 0.02 %. First we convert this percentage uncertainty into an absolute uncertainty in speed.

Since $$0.02\% = \dfrac{0.02}{100} = 0.0002,$$ the uncertainty in speed is

$$\Delta v = 0.0002 \times v = 0.0002 \times 5 \times 10^{6} = 1.0 \times 10^{3}\,{\rm m\,s^{-1}}.$$

Next, we connect this speed uncertainty to the momentum uncertainty. Momentum is defined by $$p = mv,$$ so its uncertainty is

$$\Delta p = m \,\Delta v = 9.1 \times 10^{-31}\,{\rm kg}\; \times 1.0 \times 10^{3}\,{\rm m\,s^{-1}} = 9.1 \times 10^{-28}\,{\rm kg\,m\,s^{-1}}.$$

To find the corresponding position uncertainty we invoke the Heisenberg uncertainty principle, which in its standard form reads

$$\Delta x\,\Delta p \;\ge\; \frac{h}{4\pi},$$

where $$h = 6.63 \times 10^{-34}\,{\rm J\,s}$$ and $$\pi = 3.14.$$ Assuming the minimum allowed product (equality case) gives

$$\Delta x = \frac{h}{4\pi\,\Delta p}.$$

Substituting the known numbers,

$$\Delta x = \frac{6.63 \times 10^{-34}} {4 \times 3.14 \times 9.1 \times 10^{-28}} = \frac{6.63 \times 10^{-34}} {12.56 \times 9.1 \times 10^{-28}}.$$

First multiply the constants in the denominator:

$$12.56 \times 9.1 = 114.296.$$

Hence,

$$\Delta x = \frac{6.63 \times 10^{-34}} {1.14296 \times 10^{-26}} = \left(\frac{6.63}{1.14296}\right) \times 10^{(-34) - (-26)} = 5.8 \times 10^{-8}\,{\rm m}.$$

To express this in the requested form $$x \times 10^{-9}\,{\rm m},$$ rewrite

$$5.8 \times 10^{-8}\,{\rm m} = 58 \times 10^{-9}\,{\rm m}.$$

So the nearest integer value of $$x$$ is $$58$$.

Hence, the correct answer is Option 58.