A projectile is fired in such a way that its horizontal range is equal to 2.6 times its maximum height, what is the angle of projection?

I know a similar question was asked, but I didn't understand the solution.

theta=arctan(4/2.6)

To find the angle of projection, we can use the basic equations of projectile motion. Let's break down the problem step by step:

1. A projectile is fired in such a way that its horizontal range is equal to 2.6 times its maximum height.

Let's define some variables:
- Angle of projection: θ (measured from the horizontal)
- Initial velocity: v0
- Maximum height: H
- Horizontal range: R

Now, according to the problem statement, we know that R = 2.6H.

2. First, let's find the horizontal range (R).

The horizontal range is the horizontal distance covered by the projectile before hitting the ground. It can be calculated using the formula:
R = (v0^2 * sin(2θ)) / g

Here, g is the acceleration due to gravity. However, since we are only interested in the relationship between R and H, we can ignore the value of g.

Simplifying the equation, we get:
R = (v0^2 * sin(2θ)) / g

3. Next, let's find the maximum height (H).

The maximum height is the highest point the projectile can reach in its trajectory. It can be calculated using the formula:
H = (v0^2 * sin^2(θ)) / (2g)

4. Now, we have the equations for R and H.

R = (v0^2 * sin(2θ)) / g
H = (v0^2 * sin^2(θ)) / (2g)

We are given that R = 2.6H. So, we can plug in this value in the equation and solve for θ.

2.6H = (v0^2 * sin(2θ)) / g

From here, we need to simplify the equation and solve for θ. However, the solution to this equation involves non-linear trigonometric equations, making it difficult to solve algebraically.

To solve this kind of equation, we can use numerical methods like iteration or approximation techniques. For example, with the help of a computer program or calculator, you can guess values for θ and adjust them until the left side of the equation matches the right side (2.6H).

Alternatively, you can use calculus to find the exact solution by setting up an appropriate equation and differentiating it. However, this involves more advanced math.

Hope this explanation helps!