3. Two particles X and Y have the same mass but their charges differ. If X has charge +q
and Y has charge + 9q and they accelerate in vacuum from rest through the same electric
potential difference, then their speeds vX:vY will be in the ratio:
a. 1:9
b. 9:1
c. 1:3
d. 3:1

Concept and Result

For two particles of equal mass accelerated from rest through the same potential difference,
the speed is proportional to the square root of charge, that is:

v ∝ √q

With charges +q and +9q, their speed ratio is:

v_X : v_Y = 1 : 3

Hence, the correct option is (c).

Detailed Physics Solution

When a charged particle of charge q is accelerated from rest through an electric potential difference V,
the electric potential energy lost qV is converted into kinetic energy ½mv². Thus:

qV = ½mv² ⇒ v = √(2qV / m)

Since both particles X and Y have the same mass m and move through the same potential difference V,
we can conclude:

v ∝ √q

For particle X:

q_X = +q ⇒ v_X ∝ √q

For particle Y:

q_Y = +9q ⇒ v_Y ∝ √(9q) = 3√q

Therefore, the ratio of their speeds is:

v_X : v_Y = 1 : 3

Explanation of Each Option

(a) 1 : 9

This ratio would hold if speed were directly proportional to charge (v ∝ q),
which is incorrect because speed depends on the square root of charge, not the charge itself.

(b) 9 : 1

This incorrectly suggests that the smaller‑charge particle moves much faster than the larger‑charge one,
contradicting v ∝ √q.

(c) 1 : 3 (Correct)

From v ∝ √q, the charge ratio q : 9q gives speed ratio √q : √9q = 1 : 3.
Hence, this option matches the computed result.

(d) 3 : 1

This would mean the smaller‑charge particle moves three times faster, which contradicts the formula
v = √(2qV / m). A larger charge leads to a higher speed when mass and potential difference are constant.

SEO-optimized introduction

In electrostatics and basic electromagnetism, exam problems often ask for the ratio of speeds of charged particles accelerated through the same potential difference. Using the work–energy principle and the relation between kinetic energy and electric potential energy, one can quickly derive a general formula and apply it to questions like particles with charges $+q$ and $+9q$ having equal mass.​