On a frictionless horizontal surface , assumed to be the ` x-y` plane , a small trolley `A` is moving along a straight line parallel to the `y-axis `( see figure) with a constant velocity of `(sqrt(3)-1) m//s ` . At a particular instant , when the line `OA` makes an angle of `45(@)` with the `x - axis ` , a ball is thrown along the surface from the origin `O`. Its velocity makes an angle `phi` with the x -axis and it hits the trolley .
(a) The motion of the ball is observed from the frame of the trolley . Calculate the angle `theta` made by the velocity vector of the ball with the x-axis in this frame .
(b) Find the speed of the ball with respect to the surface , if ` phi = (4 theta )//(4)`.

Answer

B, C, D

Explanation

(a) Let the ball strike the trolley at `B` . Let
` vecv_(BG) =` velocity of ball w.r.t. ground
` vecv(TG) = velocity of trolley e.r.t ground `
`:.` Velocity of ball w.r.t. trolley
` vecv(BT) = vecv(BG) - vecv(TG) `…..(i)
From triangles `OAB`
` vec(OA) + vec (AB) = vec(OB)`
`:. vec(OA) + vec v_(TG)` = vec v(BG)`
` :. vec(OA) = vecv_(BG) - vecv_(TG)` ....(ii)
From (i) and (ii) `vec(OA) = vecv _(BT)`
`rArr` velocity of ball w.r.t. trolley makes an angle of ` 45(@)` with the ` X-axis `
(b) Here ` theta = 45(@) `
`:. phi = (4 theta)/(3) = (4 xx 45)/(3) = 60(@)` ltbr gt In `DeltaOMA, ltbr gt ` theta = 45(@) rArr /_OAM = 45(@)`
`:. /_OAB = 135(@)`
Also `/_BOA = 60(@) - 45(@) = 15(@) `
Using since law in `Delta OBA`
` (v_(BG))/(sin 135(@)) = (v_(TG))/( sin 15(@)) rArr v_(BG) = 2 m//s `