Projectile motion of charged particle

The works done on the two charged particles are given by:

$W_A=F_Ay, \quad W_B=F_By$

where, $F_A$ and $F_B$ are the forces on the particles $A$ and $B$ respectively, and $y$ is the displacement due to the forces on the particles.

$F_A$ and $F_B$ are directly proportional to the respective accelerations $a_A$ and $a_B$ . And acceleration is given by $y=\frac{1}{2}at^2 \Rightarrow a = \frac{2y}{t^2}$ , where $t$ is the time taken for the particle to vertically displace $y$ . As both particles enter the uniform electric field with the same velocity, $t$ is directly proportional to the respective horizontal displacement $x$ . Therefore we have:

$\cfrac{W_A}{W_B}= \cfrac{F_A}{F_B}= \cfrac{a_A}{a_B}= \cfrac{t_B^2}{t_A^2}= \cfrac{x_B^2}{x_A^2}= \cfrac{(3d)^2}{(2d)^2}= \boxed{\cfrac{9}{4}}$