An electron and a proton are released from rest in a uniform electric field E and are found to take time `t_e` and `t_p` respectively to cover a distance x. Calculate the ratio of time taken by them. If both the particles are allowed to fall under gravity in vacuum, then calculate the ratio of time taken by them to cover a distance x starting from rest.

Explanation

We know , `x=ut+1/2at^2`
As both the particles are released from rest, u=0,
`rArr x=1/2 at^2` or `t=sqrt((2x)/a)`
For electron :
Acceleration , `a_e=F/m =(eE)/m_e`
Time taken by electron to cover a distance x , `t_e=sqrt((2x)/a_e)`
For proton :
Acceleration `a_p=F/m=(eE)/m_p`
`therefore t_p=sqrt((2x)/a_p)`
The required ratio of time period , will be
`t_e/t_p=sqrt((2x)/a_e)xxsqrt(a_p/(2x))=sqrt(a_p/a_e)`
`rArr sqrt((eE)/m_p xx m_e/(eE))=sqrt(m_e/m_p)`
`=sqrt((9.1xx10^(-31))/(1.67xx10^(-27))`
`=2.3xx10^(-2)`
When the particles fall under gravity in vacuum, the time of fall is independent of mass of body . Therefore , both the electron and proton will take the same time to cover a distance x .
Thus we conclude `t_e/t_p=1`