A particle starts with an initial velocity and passes successively over the two halves of a given distance with constant accelerations `a_1 and a_2` respectively. Show that the final velocity is the same as if the whole distance is covered with a uniform acceleration `(a_1 + a_2)/2 .`

Explanation

In the first case,

`v_(1)^(2)=u^(2)+2a_(1)s` …..(i)
`v_(2)^(2)=v_(1)^(2)+2a_(2)s` ….(ii)
Adding Eqs. (i) and (ii), we have
`v_(2)^(2)=u^(2)+2((a_(1)+a_(2))/(2))(2s)` ….(iii)
In the second case
`v^(2)=u^(2)+2((a_(1)+a_(2))/(2))(2s)` ...(iv)
From Eqs. (iii) and (iv) we can see that
`v_(2)=v`