Question #863

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.

Expert's answer

Let's write equations of the motion

for the projection which is perpendicular to the ground:

v_{y}=v_{i} sin(θ);

h=h_{0} + v_{0y}t - gt^{2}/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) = h_{0} + v_{i} sin(θ) * v_{i} sin(θ)/g - g*(v_{i} sin(θ)/g)^{2} / 2 = h_{0} + 1/2 (v_{i} sin(θ)/g)^{2}/g.

for the projection which is perpendicular to the ground:

v

h=h

v

for the projection which is parallel to the ground:

v

v

The projectile reaches maximum height when v

v

t = v

b) h(max) = h

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