Starting from rest a particle is first accelerated for time `t_1` with constant acceleration `a_1` and then stops in time `t_2` with constant retardation `a_2.` Let `v_1` be the average velocity in this case and `s_1` the total displacement. In the second case it is accelerating for the same time `t_1` with constant acceleration `2a_1` and come to rest with constant retardation `a_2` in time `t_3.` If `v_2` is the average velocity in this case and `s_2` the total displacement, then

Options

A. `v_2=2v_1`
B. `2v_1ltv_2lt4v_1`
C. `s_2=2s_1`
D. `2s_1lts_2lt4s_1`

Answer

A, D

Explanation

`a_1t_1=a_2t_2`
`v_(max)=a_1t_2=a_2t_2`
`s_1`=Area of v-t graph
`=1/2(t_1+t_2)(a_1t_1)`

`v_1=(s_1)/(t_1+t_2)=1/2a_1t_1`
`s_2=1/2(t_1+t_3)v_(max)`
`=1/2(t_1+2t_2)(2a_1t_1)`
`v_2=(s_2)/(t_1+t_3)=a_1t_1`

From the four relations we can see that
`v_2=2v_1`
and `2s_1lts_2lt4s_1`