Question

A proton, a deuteron and an alpha particle are accelerated through potentials of v , 2v and 4v respectively. Their velocities will bear a ratio

A. $1:1:1$
B. $1:\sqrt{2}:1$
C. $\sqrt{2}:1:1$
D. $1:1:\sqrt{2}$

Answer 1

The initial concept used in the question is the conversion of energy from one form to another. The energy gained by particles through moving in different potentials is converted into the kinetic energy of the particles.
A deutron has charge similar to proton and mass twice that of proton
An alpha particle has mass four times that of a proton and charges twice as that of a proton.

Complete answer:
The particles when accelerated to through different potential will gain electrostatic potential energy due to it which will be responsible for the kinetic energy and hence velocity of the particles.
Mathematically it can be written as
$\Rightarrow \dfrac{1}{2}m{{v}^{2}}=qV$,$m$ is mass of object,$v$ is velocity $q$ is charge $V$ is Potential
$\Rightarrow v=\sqrt{\dfrac{qV}{m}}$,$2$ has been removed as it will be cancelled out when taking ratios
Let the charge on proton be $x$
Then charge on deuteron will be $x$
Charge on alpha particle will be $2x$

Let the velocities be in the ratio of ${{v}_{1}},{{v}_{2}}\,and\,{{v}_{3}}$
potential through which they are accelerated is respectively $v,2v\,and\,4v$
So, for Proton ,
Let the mass be $y$
${{v}_{1}}=\sqrt{\dfrac{xv}{y}}$​
For deuteron,
Mass will be be $2y$
${{v}_{2}}=\sqrt{\dfrac{x\times 2v}{2y}}$
For alpha particle mass will be $4y$
${{v}_{3}}=\sqrt{\dfrac{2x\times 4v}{4y}}$​​
Taking the ratio all three we get
${{v}_{1}}:{{v}_{2}}:{{v}_{3}}=1:1:\sqrt{2}$

So, the correct answer is “Option D”.

Note:
Proton or hydrogen atom and deuteron are isotopes of each other
Isotopes are atomic structures of the same element with the same atomic number but different mass number.
Isobars are different chemical elements having the same atomic mass.
The particles discussed in the question are charged particles and hence are deflected by electric and magnetic fields.