Prove that the maximum horizontal range is four times the maximum height attend by a projectile when fire at an inclination so as to have maximum horizontal range
Θ = launch angle (above horizontal)
v = launch velocity
flight time = 2 [v sin(Θ)] / g
horizontal velocity = v cos(Θ)
range = horizontal velocity * flight time
... = {[v cos(Θ)] * 2 [v sin(Θ)]} / g
... = v^2 sin(2Θ) / g
max sine occurs at π/2
... so max range occurs at π/4
max range = v^2 / g
max height = average velocity * time
... = [v sin(π/4)]^2 / (2 g)
... = v^2 / (4 g)
To prove that the maximum horizontal range is four times the maximum height attained by a projectile when fired at an inclination for maximum horizontal range, we can use the equations of projectile motion.
Let's assume that the angle of inclination for the maximum horizontal range is θ.
To find the maximum horizontal range (R), we can use the formula:
R = (v^2 * sin(2θ)) / g,
where v is the initial velocity and g is the acceleration due to gravity.
To find the maximum height (H), we can use the formula:
H = (v^2 * sin^2(θ)) / (2g).
Now, let's first find the angle θ that gives the maximum horizontal range. For maximum horizontal range, the angle θ should be equal to 45 degrees (θ = 45°).
Substituting θ = 45° into the equations:
R = (v^2 * sin(2 * 45°)) / g,
H = (v^2 * sin^2(45°)) / (2g).
Using trigonometric identities, sin(90°) = 1 and sin^2(45°) = 1/2, we can simplify the equations to:
R = (v^2 * sin(90°)) / g,
H = (v^2 * (1/2)) / (2g).
Since sin(90°) = 1, the first equation becomes:
R = (v^2) / g.
Now, to prove that the maximum horizontal range is four times the maximum height, we need to show that:
R = 4H.
Substituting the equations for R and H:
(v^2) / g = 4[(v^2 * (1/2)) / (2g)].
Simplifying the equation:
(v^2) / g = (v^2) / (4g).
Canceling out (v^2) and g:
1 = 1/4.
This equation is not true, which means that the assumption that the maximum horizontal range is four times the maximum height is incorrect. Therefore, the maximum horizontal range is not four times the maximum height when fired at an inclination for maximum horizontal range.