Question

A proton is fired from far away toward the nucleus of an iron atom. Iron is element number $26$, and the diameter of the nucleus is $9.0fm$. What initial speed does the proton need to just reach the surface of the nucleus? Assume the nucleus remains at rest.

Short answer

The initial speed does the proton need to just reach the surface of the nucleus is$\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}$

Step-by-step solution

Given information and theory used 

Given :

Iron is element number :$26$

Diameter of the nucleus is : $9.0fm$

Theory used :

We can deduce from the law of conservation of energy that all kinetic energy will be converted to electric potential energy.

As a result,$\frac{m{v}^2}{2}=k\frac{q_1{q}_2}{r}$

Calculating the initial speed that the proton needs to just reach the surface of the nucleus 

We get $\frac{m{v}^2}{2}=k\frac{26·2e^2}{d}$

by substituting $q_1=26e,q_2=e,\mathrm{and}r=d/2.$

We may express the speed from here as :

$v=\sqrt{\frac{26\times 4e^{2}k}{md}}=2e\sqrt{\frac{26k}{md}}$

We can find the speed to be v2 by substituting our numerical numbers.

$$\begin{gathered} v=2\times 1.6{10}^{-19}\sqrt{\frac{26\times 9.{10}^9}{1.67\times {10}^{-27}\times 9\times {10}^{-15}}} \\ =\boxed{\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{}\mathit{m}\mathbf{/}\mathit{s}} \end{gathered}$$