Two stones are projected so as to reach the same distance from the point of projection on a horizontal surface. the maximum height reached by one exceeds the other by an amount equal to half the sum of the height attained by them. then angle of projection of the stone which attains smaller height is.
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Answer:
Let the height attained by the two stones be
h
1
&
h
2
respectively.
By given condition,
h
1
−
h
2
=
1
2
(
h
1
+
h
2
)
2
h
1
−
2
h
2
=
h
1
+
h
2
h
1
=
3
h
2
(Eq-1)
But for maximum height
h
=
u
2
sin
2
θ
2
g
But,it is also given that their ranges are equal.
R
1
=
R
2
R
=
u
2
sin
2
θ
g
⇒
u
2
1
sin
2
θ
1
g
=
u
2
2
sin
2
θ
2
g
⇒
u
2
2
u
2
1
=
sin
2
θ
1
sin
2
θ
2
(Eq-2)
Also from Eq (1)
u
2
1
sin
2
θ
1
=
3
u
2
2
sin
2
θ
2
sin
2
θ
1
=
3
u
2
2
u
2
1
sin
2
θ
2
From Eq-(2),
sin
2
θ
1
sin
2
θ
2
=
u
2
2
u
2
1
∴
sin
2
θ
1
=
3
×
sin
2
θ
1
×
sin
2
θ
2
sin
2
θ
2
∴
sin
2
θ
1
=
3
×
2
sin
θ
1
.
cos
θ
1
×
sin
2
θ
2
2
sin
θ
2
cos
θ
2
∴
sin
θ
1
cos
θ
1
=
3
sin
θ
2
cos
θ
2
∴
tan
θ
1
=
3
tan
θ
2
tan
θ
1
tan
θ
2
=
3
⇒
tan
θ
1
tan
θ
2
=
1
√
3
√
3
∴
θ
1
=
tan
−
1
(
1
√
3
)
=
30
o
θ
2
=
tan
−
1
(
√
3
)
=
60
o
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