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An aeroplane is flying in vertical plan...

An aeroplane is flying in vertical plane at an angle of `30^(@)` with the horizontal (north) and wind is is blowing from west.A package is dropped from an aeroplane. The velocity of the wind if package hits a kite flying in the space with a position vector `vec(R) = (400 sqrt(3) hat(i) + 80 hat(j) + 200 hat(k))`m with respect to the point of dropping. (Here `hat(i)` and `hat(j)` are the unit vectors along north and vertically up respectively and `hat(k)` be the unit vector due east. Assume that the bag is light enough to get carried away by the wind)

A

50 m/sec

B

25 m/sec

C

20 m/sec

D

10 m/sec

Text Solution

Verified by Experts

The correct Answer is:
B
Let velocity of the aeroplane be `vec(v)_(P) = u cos 30^(0) hat(i) + u sin 30^(0) hat(j)` and velocity of the wind be v, then
`u (sqrt3)/(2) t hat(i) + ((u)/(2) t - 5t^(2)) hat(j) + vt hat(k)`
`= 400 sqrt(3 hat(i)) + 80hat(j) + 200 hat(k)`
`rArr u(sqrt3)/(2)t = 400 sqrt3, (u)/(2) t - 5t^(2) = 80, vt = 200`
`ut = 800` and `(u)/(2)t - 5t^(2) = 80`
`rArr 400 - 5t^(2) = 80 rArr t^(2) = 64`
`rArr t = 8 sec`
`rArr v = (200)/(8) = 25 m//s`
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