How to Calculate the Range of a Projectile on a Sloped Surface?

[Newton's Protege]

I was hoping someone might be able to help with this. I have provided the problem and everything I know about solving it. Any help will be appreciated.

A projectile hits a slope at a certain point. What is the range of the projectile along the slope?

Given: initial velocity (V sub-zero), the angle alpha, the angle Beta, and g (free fall acceleration constant).

Find R (range):

Apparently, the following equation will find the range of the projectile if it hit the ground instead of a slope above the ground:

R= (initial velocity squared)(sine 2angle theta)/g

Is there a specific equation to solve for the range of a projectile when it hits a slope rather than the ground?

The equations of two dimensional motion must be used to derrive this equation. It has something to do with finding y as a function of x (y(x)). The following equation must be used: x=(initial velocity multiplied by the cosine of angle theta) multiplied by time(t). Time must be eliminated from the equation yielding time=x/inititial velocity(angle alpha+angle beta).

Thank You everyone

 

[Pyrrhus]

Start by setting up the "slope equation" (straight line), it should be of the form y = mx + b. Now recall that when the projectile hits the slope, the trajectory equation of the projectile will have the same Y as our straight line. To get the range you can use the distance equation of 2 points in a cartesian plane.

 

[kyle8921]

.

I am happy to help with this problem. The equation you have provided is correct for finding the range of a projectile when it hits the ground. However, when the projectile hits a slope, the equation will need to be modified to take into account the angle of the slope.

To find the range of a projectile when it hits a slope, we need to use the equations of two-dimensional motion, as you mentioned. In this case, we will use the equation for horizontal displacement, which is x = V sub-zero * cos(alpha) * t. We can rearrange this equation to solve for t, giving us t = x / (V sub-zero * cos(alpha)).

Now, to take into account the angle of the slope, we need to consider the vertical displacement as well. We can use the equation for vertical displacement, y = V sub-zero * sin(alpha) * t - (1/2) * g * t^2. Again, we can rearrange this equation to solve for t, giving us t = (V sub-zero * sin(alpha) ± √(V sub-zero^2 * sin(alpha)^2 + 2g * y)) / g.

We can then substitute this value of t into our equation for horizontal displacement to find the range, R, along the slope. So, our final equation for the range of a projectile hitting a slope is:

R = (V sub-zero * cos(alpha)) * (V sub-zero * sin(alpha) ± √(V sub-zero^2 * sin(alpha)^2 + 2g * y)) / g

I hope this helps you solve the problem. Remember to always double check your units and make sure they are consistent throughout the equation. Good luck!