Question

A proton is projected with velocity 7.45 $\times 10^5$ m/s towards another proton which is at rest from a very large distance. The minimum distance of approach is,
A.\[5\times {{10}^{-13}}m\]
B.\[6\times {{10}^{-4}}m\]
C.\[{{10}^{-10}}m\]
D.\[{{10}^{-8}}m\]

Answer 1

Hint: This question uses a formula of kinetic energy and electrostatic force. We will use the concept and formula of coulomb force. We will calculate the minimum distance and then equalize kinetic and electrostatic energy. After that substitute the values given in the question.
Formula used:
Kinetic energy \[=\dfrac{1}{2}m{{v}^{2}}\]
Electrostatic force \[=\dfrac{k{{e}^{2}}}{r}\]

Complete answer:
Consider a proton projected with velocity \[v\] towards another proton. Another proton is at rest and given that the distance between two protons is very large.
We know like charges repel each other. Protons are positively charged therefore force of repulsion acts between them.
Let us consider \[r\] is the minimum closest distance which means kinetic energy of proton will be equal to electrostatic energy acting between proton
\[\dfrac{1}{2}m{{v}^{2}}=\dfrac{k{{e}^{2}}}{r}\]
Here \[e\] is the charge of the electron.
Given \[v=7.45\times {{10}^{5}}\]
Rearranging the terms \[r=\dfrac{2k{{e}^{2}}}{m{{v}^{2}}}\]
\[=\dfrac{2\times 9\times {{10}^{9}}\times {{(1.6\times {{10}^{-19}})}^{2}}}{1.675\times {{10}^{-27}}\times (7.45\times {{10}^{5}})}\]
Therefore \[r=5\times {{10}^{-13}}m\]
So the minimum distance is \[5\times {{10}^{-13}}\].

Option A is correct.

Additional information:
Any object in motion possesses kinetic energy and maintains its kinetic energy unless its speed changes.
The work done to accelerate any mass of an object is referred to as kinetic energy of that object.
Coulomb force acts between two charged bodies. According to Coulomb’s law the force acting between any two charges is directly proportional to the product of charges of two bodies and inversely proportional to square of distance between the two charges.
Mass of electron \[=9.109\times {{10}^{-31}}kg\]
Mass of proton\[=1.675\times {{10}^{-27}}kg\]
Charge on electron \[=-1.6\times {{10}^{-19}}C\]
Charge on proton \[=1.6\times {{10}^{-19}}C\]

Note:
We should always remember the standard values like mass and charge of electrons, protons and neutrons. It is not always necessary that the value will be provided in the problem. We should always use the units carefully and all the parameters should be in the same system.