By Team Aakash Byju's | 15th January 2023
Check Dependency of Maximum Height in a Projectile Motion.
There are two angles of projection for which the horizontal range is the same. Show that the sum of the maximum heights for these two angles is independent of the angles of projection.
Two angles of projectile's
We know that the horizontal range in a projectile motion is given as,
Where, u is the initial velocity, α is the angle of projection, and g is the acceleration due to gravity.
R =
u sin(2α)
g
2
u =0
u =u cosθ
u =-u sinθ
u =u cosθ
u =usinθ
H
y
x
u =u cosθ
x
y
x
y
Height
x
y
m
θ
u
O
Let α and α be the two angles of projection for which the horizontal range R is the same.
1
2
R =
u sin(2α )
g
2
u sin(2α )
g
2
=
2
1
α
1
α
2
=>sin(2α )=sin(2α )
=>sin(2α )=sin(180-2α )
(∵ sin(180-2α ) is as same as sin2α
=>2α =180-2α or α =90-α or α =90- α
1
2
1
2
2
1
2
1
2
2
1
2
Let h be the maximum height of a projection with angle of projection α , then
1
1
h =
u sin α
g
2
1
2
1
Similarly,
1
2
h =
2
u sin α
g
2
2
2
u sin (90 -α )
g
2
=
=
2
u cos α
g
2
1
Thus, the sum of maximum heights is
(∵ sin α + cos α =1)
H = h +h =
u sin α
g
2
u cos α
g
2
+
1
1
2
1
2
=> H =
u (sin α + cos α )
g
2
1
1
2
2
=
u
g
2
1
1
2
2
2
It can be noted from the expression for the sum of maximum heights that angle of projection has no effect on it. Hence the sum is independent of the angles of projection.