An airplane is observed by two persons travelling at 60 km/hour in two vehicles moving in opposite directions on a straight road. To an observer in one vehicle the plane appears to cross the road track at right angles while to the observer in the other vehicle the angle appears to be `45^(@)` . At what angle does the plane actually cross the road track and what is its speed relative to ground?

Answer

`theta=tan^(-1)2;v=134.16` km/hour

Explanation

Let the initial position vector is `x'hati+y'hatj+z'hatk=vecr_(A)`.
Now, `a_(X)=(12)/(3), a_(y)=-(3)/(3),a_(z)=(21-10)/(3)`
`vecr_(B)=15hati+7hatj-6hatk`
Now, `15-x'=u_(x)xx2+(1)/(2)xx4xx(2)^(2)`
`7-y'=u_(y)xx2+(1)/(2)(-1)xx(2)^(2)`
`-6-z=u_(z)xx2+(1)/(2)xx(11)/(3)xx(2)^(2)`
`vecv_(B)=12hati+hatj=4hatk`
`vecv_(B)=v_(x)hati+v_(y)hatj+v_(z)hatk`
`(v_(x)-u_(x))/(a_(x)) (u_(y)-u_(y))/(a_(y))=t,(v_(z)-u_(2))/(a_(x))=t`