Finding the Angle in Projectile Motion

[Robotics 1764]

Homework Statement

Hello, I am attempting to derive an equation for some pretty specific needs. The equation I need is one that solves for the angle needed to launch a projectile at a given velocity to an object at a given distance and height different from launcher. The catch is that I would like to include the acceleration due to air resistance (which I can find later and input) as an acceleration in the x direction.

The goal is to solve for theta (I will use an o)

Homework Equations

The only variables that can be present in the equation are:
ax, ay (gravity), x, y, v (velocity launched at, actually, it is the speed, I don't know the angle)

The equations I have been attempting to use are:

y = vy*t + (g/2)*t^2
x = vx*t + (a/2)*t^2
v = sqrt(vx^2 + vy^2)
sin o = vy/v
cos o = vx/v

The Attempt at a Solution

I keep trying to derive but keep ending up circling (deriving until I accidentally reach a variable I tried to get rid of earlier). Does anyone have an equation already derived they could show me that they could also explain how they derived, or know how I could go about deriving my goal equation successfully? Or any other formulae I should be using?

Thanks.

 

[tiny-tim]

Welcome to PF!

The equation I need is one that solves for the angle needed to launch a projectile at a given velocity to an object at a given distance and height different from launcher.

The catch is that I would like to include the acceleration due to air resistance (which I can find later and input) as an acceleration in the x direction.
…
The equations I have been attempting to use are:

y = vy*t + (g/2)*t^2
x = vx*t + (a/2)*t^2
v = sqrt(vx^2 + vy^2)
sin o = vy/v
cos o = vx/v

Hi Robotics 1764! Welcome to PF!

(air resistance would never be like that, but anyway …)

Hint: since you have constant acceleration in both the x and y directions, try combining them (accelerations obey the vector law of addition, of course) into a single acceleration, and then use coordinates parallel and perpendicular to that direction.

 

[thomate1]

I understand your need to derive an equation for your specific needs in projectile motion. From your given variables, it seems like you are looking for an equation that takes into account both the acceleration due to gravity and the acceleration due to air resistance in the x direction. This can be a complex problem, but I can offer some guidance and potential solutions.

Firstly, it is important to note that the equations you have been attempting to use are for projectile motion in the absence of air resistance. To incorporate air resistance, we need to consider the force of air resistance, which is often modeled as being proportional to the square of the velocity (Fres = -bv^2, where b is a constant). This force will act in the opposite direction to the motion of the projectile, so it will affect both the x and y components of the motion.

One approach to solving this problem is to use vector notation and break down the forces acting on the projectile into x and y components. The x component will include the force of air resistance, while the y component will include both the force of gravity and the normal force (which cancels out the force of gravity in the y direction). From there, you can use the equations of motion to solve for the angle and velocity needed to reach your desired distance and height.

Another approach is to use numerical methods, such as computer simulations, to solve for the angle and velocity. This can be a more accurate and efficient method, as it takes into account the changing forces and velocities throughout the trajectory.

I hope this provides some helpful insight and guidance. It is always important to carefully consider all variables and forces acting on a system when trying to derive equations for specific needs. Good luck with your research!