A particle of rest mass `m_0` starts moving at a moment `t=0` due to a constant force F. Find the time dependence of the particle's velocity and of the distance covered.

Explanation

The equation of motion is
`(d)/(dt)((m_0v)/(sqrt(1-v^2/c^2)))=F`
Integrating `=(v//c)/(sqrt(1-v^2/c^2))=(beta)/(sqrt(1-beta^2))=(F_t)/(m_0c)` using `v=0` for `t=0`
`(beta^2)/(1-beta^2)=((Ft)/(m_0c))^2` or, `beta^2=((Ft)^2)/((Ft)^2+(m_0c)^2)` or, `v=(Fct)/(sqrt((m_0c)^2+(Ft)^2))`
or `x=int (Fctdt)/(sqrt(F^2t^2+m_0^2c^2))=c/Fint(xidxi)/(sqrt(xi^2(m_0c)^2))=c/Fsqrt(F^2t^2+m_0^2c^2)+const ant`
or using `x=0` at `t=0`, we get, `x=sqrt(c^2t^2+((m_0c^2)/(F))^2)-(m_0c^2)/(F)`