A proton and an $\alpha -$ particle start from rest in a uniform electric field, then the ratio of times of flight to travel same distance in the field is

Given data: A proton and an α- particle

Concept used: Electric fields and force

Detailed solution: 

Let m₁ and m₂ be the mass of the proton and α- particle

q₁ and q₂ be the charge of the proton and α- particle

We know that force acting on the particle placed in the electric field is 

$$\begin{gathered} F=Eq \\ ma=qE \\ a=\left(\frac{q}{m}\right)E \end{gathered}$$

From equations of motion, we have

$s=ut+\frac{1}{2}a{t}^2$

Since both particles starts from rest u=0

$$\begin{gathered} s=\frac{1}{2}a{t}^2 \\ {s}_1=s_2 \\ \frac{1}{2}{a}_1{t}_1^2=\frac{1}{2}{a}_2{t}_2^2 \\ \frac{1}{2}\left(\frac{q_1}{m_1}\right)E{t}_1^2=\frac{1}{2}\left(\frac{q_2}{m_2}\right)E{t}_2^2 \\ \frac{q_1}{m_1}{t}_1^2=\frac{q_2}{m_2}{t}_2^2 \end{gathered}$$

In case of α- particle m₂=4m₁, q₂=2q₁

$$\begin{gathered} t_1^2=\frac{1}{2}\times t_2^2 \\ \frac{t_1}{t_2}=\frac{1}{\sqrt{2}} \\ {t}_1:t_2=1:\sqrt{2} \end{gathered}$$