How Does Projectile Angle Affect Range When Fired Up a Sloped Hill?

[Serenity16]

Homework Statement

A projectile is fired with initial speed v0 at an elevation angle (alpha) up a hill of slope (beta) (alpha > beta).

a) How far up the hill will the projectile land?
b) At what angle (alpha) will the range be a maximum?
c) What is the maximum range?

Homework Equations

x = v0tcos(alpha)
y = -(gt^2)/2 + v0sin(alpha)
r = (x^2 + y^2)^(1/2)

The Attempt at a Solution

I had thought that the solution to this problem might be as simple as finding where the line that represents the hill intersects with the line that represents the trajectory of the projectile. But I'm not sure how to get there from here. I would know how to solve this if the range I was concerned about was the range from where the projectile is launched to where it hits the ground again (i.e. y2 = 0), but this has me stumped. Anything to point me in the right direction would be appreciated!

 

[jaseh86]

Hmm.. in the time I've had to think about this, I could only see one way to approach this, and it probably isn't the simplest approach.

As you said, you need to find the intersection between the projectile path and the line that represent the hill. A projectile takes the path of a parabola, while the hill would be a linear line. Do you know how to find the cartesian equation of a parabolic projectile path? (hint: substitute the t's so you only have y and x).

If you're stumped about the hill, just remember how to find the gradient of a line given its angle. Once you have TWO equations, it should just be a matter of algebra.

 

[baba89]

I would approach this problem by first breaking it down into smaller components and applying the relevant equations.

a) To determine how far up the hill the projectile will land, we can use the equation for horizontal displacement, x = v0tcos(alpha). We know that at the point of impact, the vertical displacement (y) will be equal to the elevation of the hill (beta). So, we can set y = beta and solve for t. This will give us the time it takes for the projectile to reach the top of the hill. Then, we can plug this value of t into the equation for horizontal displacement to find the distance x.

b) To find the angle (alpha) at which the range will be maximum, we can use the equation for range, r = (x^2 + y^2)^(1/2). We can differentiate this equation with respect to alpha and set it equal to 0 to find the angle at which the derivative is 0. Solving for alpha will give us the angle at which the range is maximum.

c) Finally, to find the maximum range, we can use the equation for range and plug in the value of alpha we found in part b. This will give us the maximum value of r.

It is important to note that the equations used here assume ideal projectile motion, neglecting air resistance and other external forces. In reality, these factors would affect the projectile's trajectory and the results may differ. Additionally, the slope of the hill may also have an impact on the projectile's motion, and further analysis may be needed to account for this.