8. An object A is kept fixed at the point x = 3 m and y = 1.25 m on a plank P raised above the ground. At time t = 0 the plank starts moving along the + xdirection with an acceleration 1.5 m/s2. At the same instant a stone is projected from the origin with a velocity vector u as shown. A stationary person on the ground observes the stone hitting the object during its downward motion at an angle of 45o to the horizontal. All the motions are in X-Y plane. Find vector u and the time after which the stone hits the object. Take g = 10 m/s2.

dear student,

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It is a big question but the answer is rather easy

see rel vel of ball w.r.t trolley,

v(rel)=vcosψ i+ v sinψ j -(root{3}-1)j

say if some φ anghle is made by this vector 

so tan φ = v sinψ - ( root3-1)/vcosψ.    ........(1)

as the ball hits the trolley so the relative velocity is towards trolley; as a result the angle made by vel vector of ball w.r.t trolley is 45 with +ve X-axis

if ψ=4Θ/3, keep this value in(1). at t=0 & φ= 45  solve the equation and get the value of v which is 2m/s at 60 degree to X-axis