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On a frictionless horizontal surface , ...

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)`.

Text Solution

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The correct Answer is:
2
`overset(-)A+overset(-)B=a [(1+cos omegat)hati+sin omega thatj], overset(-)A-overset(-)B=a[(1-cos omega t)hati-sin omegat hatj]`
`|overset(-)A+overset(-)B|=a|2 cos ((omegat)/(2))|, |overset(-)A-overset(-)B|=a |2 sin ((omegat)/(2))|`
`|(overset(-)A+overset(-)B)/(A-B)|=|cot ((omegat)/(2))|=sqrt3 rArr (omegat)/(2)=pi/6 rArr t=2`
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