A cannon shell is fired to hit a target at a horizontal distance `d`. However , it breaks into two parts of mass ratio `1 : 2` at its height point. The smaller part returns to cannon. The other part
(i) will fall at a distance `d//2` beyond the target
(ii) have eight times the kinetic energy of smaller part
(iii) the increase in kinetic energy of system after explosion is `200 %`
(iv) after explosion , heavier part will take more time as compared to smaller part , to strike the ground

Options

A. `(i) ,(ii)`
B. `(i) ,(ii),(iii)`
C. `(i) ,(iii)`
D. all options are correct

Answer

B

Explanation

`3 md = m xx 0 + 2m x_(0) rArr x_(0) = (3d)/(2)`
`x' = x_(0) - d = (3d)/(2) - d = (d)/(2)`
(i) is `O.K`.
Momentum conservation at the highest point is
`3m v cos theta = - mv cos theta + 2mv' rArr v' = 2v cos theta`
`(K_(2m))/(K_(m)) = ((1)/(2) ( 2 v cos theta)^(2))/((1)/(2) xx m ( v cos theta)^(2)) = 8`
(ii) is `O.K`.
`K.E.` before explosion
`K_(i) = (1)/(2) xx 3m( v cos theta)^(2) = (3m)/(2) v^(2) cos^(2) theta`
`K.E.` after explosion
`K_(f) = (1)/(2) m(v cos theta)^(2) + (1)/(2) xx 2m ( 2v cos theta)^(2)`
` = (9m v^(2) cos^(2) theta)/(2)`
`((K_(f))/(K_(i)) - 1) xx 100 = 200%`
(iii) is `O.K.`
Both parts will strike the ground simultaneously , (iv) is wrong.