I've been studying multivariable calculus the last 2 weeks, and I understand (I think) how to optimize 2 variable equations through normal optimization and constrained optimization via Lagrange.
I couldn't draw the connection though when I tried to optimize the projectile motion equations, attempting to find the optimal angle to cover a constant distance in the minimum amount of time.
$$π₯ = π£\cos(π)π‘$$ $$π¦ = π£\sin(π)t β \frac{1}{2}π\mathrm{π‘}^2$$
I also don't understand how to implement the constrain of covering a certain distance nor how to rewrite this equation minimizing time in the optimization. I am genuinely confused by the amount of variables and can't seem to 'make the leap' so to speak from what I know to what I don't.
Edit:
I apologize for the lack of clarity on this post.
The problem is covering a certain distance in the least amount of time by finding the optimal launch angle at a constant velocity.
Through your responses, I think this means that $d$, total displacement, and $v$, constant velocity have been defined in our equations. This leaves θ and $t$ as our two variables.
If we then rewrite the two main equations in terms of $t$ as a function of θ, I believe this would point us towards finding the solution.
$$t = \frac{x}{vcos(π)}$$ and $$ t = \frac{y}{vsin(π)-\frac{1}{2}gt}$$ This would mean $$ \frac{x}{vcos(π)} = \frac{y}{vsin(π)-\frac{1}{2}gt}$$
I am stumped however on what to do after this. Mainly, given a total displacement of $d$, how would I go about writing that in terms of $x$ and $y$. I think $x = dcos(π)$ and $y = dsin(π)$ through vector resolution, but how would I incorporate that into the equations to calculate the optimal angle for the least amount of time.
If anything is incorrect or still unclear, do let me know. I appreciate all of your responses; they have indeed helped expand my thinking about this.