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My background

I'm working on a project for a client in the gaming industry. Newtonian physics and vector math is usually my forte, but I'm usually solving problems in directions opposite of what this one requires, and I hit a snag that I'm trying to work through. I'm sure there's a relatively simple answer, but I haven't been able to find one here on SE anywhere that will work with my problem.

There was a similar question asked, but the formulas used by both the answerer and the OP use angular notation for its velocity, not vector notation, and as far as I can tell the answer does not cover the same kind of problem I'm trying to solve.

My problem

I've made a formula that will calculate the individual $x$ and $y$ velocities needed to launch a projectile from point $\vec{\mathit{p1}}$ so that it will reach a height of $\mathit{h}$ and land at point $\vec{\mathit{p2}}$.

It works when $\vec{p1}_y$ and $\vec{p2}_y$ are equal...

It works when p1_y and p2_y are equal...

but starts to miss the farther away $p2_y$ gets from $p1_y$.

but starts to miss the farther away p2_y gets from p1_y

I believe this is because my $x$ velocity calculation assumes the projectile will travel the entire distance of the arc to $\vec{p2}$, but ends up falling further/shorter when $\vec{p2}$ is higher or lower than $\vec{p1}$. Therefore, I'm trying to find a way to compensate for the $y$ displacement of $\vec{p2}_y$ in my $x$ velocity calculations

Here's what I have so far: $$ points: \vec{p1},\space \vec{p2} $$ $$ g = 9.8m/s $$ $$ t=\text{Time. Equivalent to $f(x)=x$.} $$ $$ \large{\vec{v_0}=[}f_{\small{vx}}(\small{\frac{v_y}{g}}\large{)},\space v_y\large{]} $$

I first calculate the $y$ velocity I need to reach height $h$.

$$ v_y=\sqrt{2g\space\cdot(h-\vec{p1}_{\large{y}})} $$

With the $y$ velocity, I can now use the constant force of gravity to calculate the air time. $$ t=\frac{v_y}{g} $$

I then plug the air time into a function that finds the $x$ velocity needed to reach $p2_{\large{x}}$ before my $y$ velocity runs out. $$ f_{vx}(t)=\frac{\frac{1}{2}(\vec{p2}_x - \vec{p1}_x)}{t} $$ (If the $x$ displacement is not halved in the above formula, the peak of the arc will be directly over $\vec{p2}$)

This results in the equation for $\vec{v_0}$.

What I've tried so far

I tried finding a way to use a proportion of the untraveled height to adjust the air time that $f_{vx}()$ uses. The way I tried was this:

$$ c_d=\frac{h-(p2_y-p1_y)}{h}+1 $$ (The $+1$ is added because the arc is assumed to always have traveled at least $\frac{1}{2}f_{vx}(v_y)$, therefore the extra amount over 1 is what is being added on to the $x$ velocity)

After I got this correctional value, I tried inserting it into many different spots in the equation, but none seemed to produce the desired result. Am I on the right track?

I feel like I'm close, but missing something important. Any help that can be given would be very appreciated!

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  • $\begingroup$ I think it might help you to set $x_0=0$ and $y_0=0$ (without loss of generality since only the distance differences matter). In which case, you know that $v_{0y}$ must equal $\sqrt{2gh}$. They you can use $y_f = \frac{-gx_f^2}{2v_{0x}^2}+\frac{x_f v_{0y}}{v_{0x}}$ to solve for $v_{0x}$ given the final point. $\endgroup$
    – hft
    Commented Oct 6, 2023 at 0:53

1 Answer 1

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The height, $h$, determines the $y$-component of the initial velocity.

$$y_{\text{max}} = h \;\;\;\;\Rightarrow\;\;\;\; v_y = \sqrt{2g(h-P_{1y})}$$

The end points, $P_1$ and $P_2$, constrain the trajectory such that $$\begin{equation} \begin{cases} P_{2x} &= P_{1x} + v_xT \\ P_{2y} &= P_{1y} + v_yT-\frac{1}{2}\,g\,T^2 \\ \end{cases} \end{equation}$$

$T$ is the time after launch when the particle reaches point $P_2$. This is a system with two equations and two unknowns, $v_x$ and $T$. Solving for $v_x$, there are two solutions.

$$\begin{align} v_x = \sqrt{\frac{g}{2(h-P_{1y})}}\frac{P_{2x}-P_{1x}}{1\pm\sqrt{\frac{h-P_{2y}}{h-P_{1y}}}} \end{align}$$

The (+) solution has the trajectory reach the maximum height, $h$, before reaching point $P_2$, and the (-) solution's trajectory reaches the maximum height after passing through $P_2$.

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  • $\begingroup$ YES! This is it! Thank you so much for your help! $\endgroup$
    – PlugN'Play
    Commented Oct 6, 2023 at 2:49

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