Prove that the maximum horizontal range is 4times the max height attaind by projectile when fired at an inclined so as to have max. Horizontal range
To prove that the maximum horizontal range is 4 times the maximum height attained by a projectile when fired at an inclined angle for maximum horizontal range, we need to use some principles of physics and equations of projectile motion.
Let's consider a projectile launched at an angle θ with an initial velocity v₀. The vertical and horizontal components of the initial velocity can be calculated using the following equations:
Vertical component (v₀y) = v₀ * sin(θ)
Horizontal component (v₀x) = v₀ * cos(θ)
The time of flight of the projectile can be determined using the equation:
Time of flight (t) = 2 * v₀y / g
where g is the acceleration due to gravity.
The maximum height attained by the projectile can be calculated using the equation:
Maximum height (H) = (v₀y)² / (2 * g)
Next, let's find the horizontal range (R) of the projectile. The horizontal range is the total distance traveled by the projectile horizontally before hitting the ground. The horizontal range can be calculated using the equation:
Horizontal range (R) = v₀x * t
Substituting the values from earlier equations:
R = (v₀ * cos(θ)) * (2 * v₀ * sin(θ) / g)
Using trigonometric identity (2sinθcosθ = sin2θ):
R = (v₀² * sin(2θ)) / g
Now we need to find the angle θ that maximizes the horizontal range. To do this, we differentiate R with respect to θ and equate it to zero to find the critical angle:
dR/dθ = (2v₀²cos(2θ)) / g = 0
cos(2θ) = 0
2θ = π/2
θ = π/4
Hence, the angle θ for maximum range is π/4 radians (45 degrees).
Now, substituting θ = π/4 in the equation for horizontal range R, we get:
R = (v₀² * sin(2(π/4))) / g
R = (v₀² * sin(π/2)) / g
R = (v₀²) / g
So, when fired at an inclined angle to achieve the maximum horizontal range, R = (v₀²) / g.
To determine the maximum height H, we substitute θ = π/4 in the equation for maximum height:
H = (v₀ * sin(π/4))² / (2 * g)
H = (v₀ / √2)² / (2 * g)
H = (v₀²) / (4 * g)
We've found that the maximum horizontal range R = (v₀²) / g and the maximum height H = (v₀²) / (4 * g).
To prove that the maximum horizontal range is 4 times the maximum height, we divide the two equations:
R / H = [(v₀²) / g] / [(v₀²) / (4 * g)]
R / H = 4
Therefore, the maximum horizontal range is indeed 4 times the maximum height attained by a projectile when it is fired at an inclined angle to achieve the maximum horizontal range.