prove that the maximum horizontal range is 4 times the maximum height attained by a projectile which is fired long the required oblique direction
The maximum range is achieved when θ=π/4
R = v^2 sin2θ/g
h = v^2 sin^2 θ / 2g
R/h = 4cotθ
at θ=π/4, R/h = 4
To prove that the maximum horizontal range of a projectile is four times the maximum height attained, we need to analyze the projectile motion and derive its equations.
First, let's define some variables:
- Horizontal range (R): the distance covered horizontally by the projectile.
- Maximum height (H): the maximum vertical distance reached by the projectile.
- Launch angle (θ): the angle at which the projectile is launched with respect to the horizontal.
- Initial velocity (V): the magnitude of the projectile's initial velocity.
Now, let's derive the equations for the horizontal range and the maximum height.
1. Horizontal Range (R):
The horizontal range can be calculated using the formula:
R = (V^2 * sin(2θ)) / g,
where g is the acceleration due to gravity.
2. Maximum Height (H):
The maximum height can be calculated using the formula:
H = (V^2 * sin^2(θ)) / (2g).
To prove that R is four times H, we need to show that R = 4H.
Substituting the equation for R into the equation for H, we get:
H = (V^2 * sin^2(θ)) / (2g).
R = (V^2 * sin(2θ)) / g.
Let's calculate R/H:
(R / H) = [(V^2 * sin(2θ)) / g] / [(V^2 * sin^2(θ)) / (2g)].
(R / H) = 2 * (sin(2θ) / sin^2(θ)).
Using the double-angle formula, sin(2θ) = 2 * sin(θ) * cos(θ), we can simplify the expression further:
(R / H) = 2 * [(2 * sin(θ) * cos(θ)) / sin^2(θ)].
Let's simplify the expression inside the brackets:
(R / H) = 2 * [(2 * sin(θ) * cos(θ)) / (1 - cos^2(θ))].
(R / H) = 4 * (sin(θ) / (1 - cos^2(θ))).
Using the identity sin^2(θ) + cos^2(θ) = 1, we can rewrite the expression:
(R / H) = 4 * (sin(θ) / sin^2(θ)).
(R / H) = 4 / sin(θ).
Since sin(θ) cannot be zero (as it represents the launch angle), we can conclude that:
(R / H) = 4.
Therefore, we have proved that the maximum horizontal range is four times the maximum height attained by a projectile fired along the required oblique direction.
To prove that the maximum horizontal range is four times the maximum height attained by a projectile fired along the required oblique direction, we can use the equations of motion.
Let's assume that the initial velocity of the projectile is V₀ and it is fired at an angle θ with respect to the horizontal direction.
The horizontal range (R) and the maximum height (H) can be calculated as follows:
1. Horizontal Range (R):
The horizontal component of the initial velocity (V₀x) is V₀x = V₀ * cos(θ).
The time it takes for the projectile to reach the maximum height, and then return to the ground, is given by:
t = (2 * V₀ * sin(θ)) / g,
where g is the acceleration due to gravity.
Using this time in the horizontal range formula:
R = V₀x * t
= (V₀ * cos(θ)) * (2 * V₀ * sin(θ)) / g
= (2 * V₀² * cos(θ) * sin(θ)) / g
2. Maximum Height (H):
The vertical component of the initial velocity (V₀y) is V₀y = V₀ * sin(θ).
The time it takes for the projectile to reach the maximum height is given by:
t_max = V₀y / g.
Using this time in the maximum height formula:
H = (V₀y)² / (2 * g)
= (V₀ * sin(θ))² / (2 * g)
= (V₀² * sin²(θ)) / (2 * g)
Now, let's compare the ratio of R and H:
R / H = [(2 * V₀² * cos(θ) * sin(θ)) / g] / [(V₀² * sin²(θ)) / (2 * g)]
= (2 * V₀² * cos(θ) * sin(θ)) / (V₀² * sin²(θ))
= 2 * cos(θ) / sin(θ)
= 2 * (cos(θ) / sin(θ))
= 2 * (cot(θ))
Since cot(θ) = 1 / tan(θ), we can simplify further:
R / H = 2 * (1 / tan(θ))
= 2 * (tan(θ))⁻¹
Using the trigonometric identity tan(θ) = sin(θ) / cos(θ), we get:
R / H = 2 * (cos(θ) / sin(θ)) = 2 * (cot(θ))
Since cot(θ) is the reciprocal of tan(θ), it can be written as 1 / tan(θ). Therefore:
R / H = 2 * (1 / tan(θ))
= 2 / tan(θ)
We know that tan(θ) = H / R, so we can substitute that in:
R / H = 2 / (H / R)
= (2 * R) / H
Therefore, R / H = (2 * R) / H
Multiplying both sides by H:
R = 2 * R
So, we can see that the maximum horizontal range R is indeed equal to 2 times R.
Hence, the maximum horizontal range is four times the maximum height attained by a projectile fired along the required oblique direction.