Following are four different relations about displacement, velocity and acceleration for the motion of a particle in general. Choose the incorrect one (s)

Options

A. `v_("av")=(1)/(2)[v(t_(1))+v(t_(2))]`
B. `v_("av")=(r(t_(2))-r(t_(1)))/(t_(2)-t_(1))`
C. `r=(1)/(2)(v(t_(2))-v(t_(1))) (t_(2)-t_(1))`
D. `a_("av")=(v(t_(2))-v(t_(1)))/(t_(2)-t_(1))`

Answer

A, C

Explanation

If an object undergoes a displacement `Delta r` in time `Delta t`, its average velocity is given by
`v=(Deltar)/(Delta t)=(r_(2)-r_(1))/(t_(2)-t_(1))`, where `r_(1)` and `r_(2)` are position vectors corresponding to time `t_(1)` and `t_(2)`.
It the velocity of an object changes from `v_(1)` to `v_(2)` in time `Delta t`. Average acceleration is given by
`a_(av)=(Deltav)/(Delta t)=(v_(2)-v_(1))/(t_(2)-t_(2))`
But , when acceleration is non-uniform
`v_("av")!= (v_(1)+v_(2))/(2)`
We can write `" " Delta v=(Delta r)/(Delta t)`
Hence, `" " Delta r=r_(2)-r_(1)=(v_(2)-v_(1))(t_(2)-t_(1))`