Question

If r is the range of a projectile on a horizontal plane and x is a maximum height then maximum horizontal range with the same velocity of projection is

Answer 1

consider a tower AB of height x .let a body be projected from its top with a velocity u in the horizontal direction.
under the combined effect of uniform velocity u in the horizontal direction and the force of gravity in the vertical down ward direction the body moves along a path AC and falls after time t at C on the ground .
let the horizontal range BC be r
in vertical down ward direction,
initial velocity u=0
acceleration due to the gravity g=+g
vertical displacement S =x
time to travel t=t
substitute in the above value of equation
S=ut +1/2at (square)
we get x=0t+ 1/2gt(square)
x=1/2gt(square) .....1equation
since the velocity in the horizontal direction is uniform,
horizontal displacement =horizontal velocity ×time
r=ut
t=x/u......2equation
substituting the value of t in equation 1
x =1/2gt(square)
x=1/2g(r/u)square
x=1/2g(r/u)square
in the above expression g/2u(square), let it be A
x=Ar(square)
this equation resembles the equation of a parabola.hence the path in the projectile travels is a parabola.