A point moves with a uniform acceleration and ` v_1 ,v_2, v_3` denote the average velocities in the three successive intervals of time `t _1` ,` t_2` and `t_3`. Find the ratio of ( `v_1` - `v_2`) and ( `v_2` - `v_3`).

Options

A. `t_1 - t_2/t_2 - t_3`
B. `t_1 - t_3/t_2 + t_3`
C. `t_1 + t_2/t_2 - t_3`
D. `t_1 + t_2/t_2 + t_3`

Answer

D

Explanation

Let u be initial velocity and a be uniform acceleration.
Average velocities in the intervals from 0 to `t_1, t_1` to `t_2` and `t_2` to `t_3` are
`v_1 = u + U + at_1/2 = u + a/2 t_1` …..(i)
`v_2 = u + at_1 + u + a(t_1 + t_2) = u + at_1 + q/2 t_2` ...(ii)
`v_3 = u + a(t_1 + t_2) + u + a(t_1 + t_2 + t_3)/2 = u + at_1 + at_2 + a/2 t_3` ... (iii)
Subtract (i) from (ii) , we get
`v_2 - v_1 = a/2 (t_1 + t_2)`
Subtract (ii) from (iii), we get
`v_3 - v_2 = a/2 (t_2 + t_3)` ..(v)
Divide (iv) by (v), we get
`v_2 - v_1 / v_3 - v_2 = (t_1 + t_2)/(t_2 + t_3) ` or `v_1 - v_2/v_2 - v_3 = (t_1 + t_2)/(t_2 + t_3)`