Two hunters simultaneously aim their guns at a bird sitting on a tree. The first hunter fires his shot with velocity 117·6m/s at angle 30°. The second hunter ahead of first hunter by a distance 19·6√3 m fires his shot at angle 60° and both the shots hit the bird simultaneously. Find the speed of fire of shot by second hunter, the distance of foot of tree from first hunter and height of bird. from Physics Motion in A Plane Class 11 Manipur Board
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Physics Part I

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Two hunters simultaneously aim their guns at a bird sitting on a tree. The first hunter fires his shot with velocity 117·6m/s at angle 30°. The second hunter ahead of first hunter by a distance 19·6√3 m fires his shot at angle 60° and both the shots hit the bird simultaneously. Find the speed of fire of shot by second hunter, the distance of foot of tree from first hunter and height of bird.

The situation is as shown in figure.



A is the point of first shot. 

With respect to A, the position of bird is straight P left right arrow left parenthesis straight x comma space straight y right parenthesis.

Position of first shot at any instant is, 

straight x subscript 1 space equals space straight u subscript 1 space cosθ subscript 1 straight t space
space space space space equals space 117.6 space cos space 30 to the power of straight o space straight t space

space space space space equals space 58.8 square root of 3 space straight t space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space

straight y subscript 1 space equals space straight u subscript 1 sinθ subscript 1 straight t space minus space 1 half gt squared space
space space space space space equals space 117.6 space sin space 30 to the power of straight o space straight t space minus 1 half 9.8 space straight t squared space
space space space space space equals space 58.8 space straight t space minus space 4.9 straight t squared space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space

Position space of space second space shot space at space any space instant space is

straight x subscript 2 space equals space straight x subscript infinity space plus space straight u subscript 2 cosθ subscript 2 space straight t space
space
space space space space equals space 19.6 square root of 3 space plus space straight u space cos 60 to the power of straight o straight t space

space space space space equals space 19.6 space square root of 3 space plus straight u over 2 straight t space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis

straight y subscript 2 space equals space straight u subscript 2 sin space straight theta subscript 2 straight t space minus space 1 half gt squared space
space space space space equals space straight u space sin space 60 to the power of 0 space straight t space minus 1 half 9.8 straight t squared space
space space space space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t space minus space 4.9 space straight t squared space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 4 right parenthesis

As both the shots hit the bird simultaneously. Therefore at the instant of hitting, positions of both the shots will be same. 

i.e. straight x subscript 1 equals straight x subscript 2 space space space and space space straight y subscript 1 equals straight y subscript 2 

From condition straight y subscript 1 equals straight y subscript 2 comma we get 

58.8 space straight t space minus space 4.9 space straight t squared space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t space minus space 4.9 straight t squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space
space space space space space space space space space space space space space space space space 58.8 space straight t space equals space fraction numerator straight u square root of 3 over denominator 2 end fraction straight t
space space space space space straight u space equals space fraction numerator 117.6 over denominator square root of 3 end fraction space equals space 39.2 space square root of 3 space straight m divided by straight s space

From space condition space straight x subscript 1 space equals space straight x subscript 2 comma space we space get
space space space space space space 58.8 square root of 3 straight t space equals space 19.6 square root of 3 space plus space straight u over 2 straight t space
rightwards double arrow space space space 58.8 square root of 3 straight t space equals space 19.6 square root of 3 space plus space 19.6 square root of 3 straight t space

rightwards double arrow space space space space space space space straight t space equals space 1 half sec

The space distance space of space foot space of space tree space from space the space first space hunter space is comma space

straight S space equals space straight x subscript 1 space equals space straight u subscript 1 cosθ subscript 1 straight t space equals space 117.6 space cos 30 to the power of straight o cross times space 1 half
space space space space equals space 29.4 square root of 3 space straight m space

The space height space of space the space bird space is comma space

straight h space equals space straight y subscript 1 space equals space 58.8 space straight t space minus space 4.9 straight t squared space
space space space space space space space space space space space space space space equals 58.8 cross times 1 half space minus 4.9 open parentheses 1 half close parentheses squared space
space space space space space space space space space space space space space space space equals space 28.175 space straight m

              

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