Answer to Question #863 in Mechanics | Relativity for duha

Question

A projectile is fired from the top of a cliff of height h above the ocean below. The projectile is fired at an angle θ above the horizontal and with an initial speed v[sub]i[/sub].(a) Find a symbolic expression in terms of the variables v[sub]i[/sub], g, and θ for the time at which the projectile reaches its maximum height.(b) Using the result of part (a), find an expression for the maximum height hmax above the ocean attained by the projectile in terms of h, v[sub]i[/sub], g, and θ.

Answer 1

Let's write equations of the motion
for the projection which is perpendicular to the ground:
v_y=v_i sin(θ);
h=h₀ + v_0yt - gt²/2;
v_y = v_0y - gt.
for the projection which is parallel to the ground:
v_x=v_i cos(θ);
v_x=v_0x.
The projectile reaches maximum height when v_y = 0
v_0y - gt = 0;
t = v_0y / g = v_i sin(θ) / g.
b) h(max) = h₀ + v_i sin(θ) * v_i sin(θ)/g - g*(v_i sin(θ)/g)² / 2 = h₀ + 1/2 (v_i sin(θ)/g)²/g.