godot Obstacle-climbing Olympics 2024-11-12How fast do you think you can climb up something? Assuming a scene with only the ground, an elevated ground, the sky, and you?
First let’s assume that once someone jumped they follow projectile motion equations along the Y axis purely, meaning that once we’ve been set in motion the only force applied is gravity pulling down:
With \(\theta = 90^{\circ}\) we have \(\sin(90^{\circ}) = 1\) and:
\[\begin{aligned} \text{Position (m) } \quad y &= v_{0}t-\frac{1}{2}gt^2 \\ \text{Velocity (ms}^{-1}\text{) } \quad v_{y} &= v_{0}-gt \\ \text{Acceleration (ms}^{-2}\text{) } \quad a_{y} &= -g \end{aligned}\]Here we’re interested in \(v_{0}\) as it’s the velocity we got from jumping. Once in the air, this value will be constant in our system of equations.
What we’re looking for is for what value of \(v_{0}\) will we eventually reach some apex \(y_{apex}\)?
Let’s consider velocity: Before reaching the apex we should be climbing, after passing the apex we should be falling, and our velocity should be null at the apex:
\[0 = v_{0}-gt_{apex} = v_{apex}\]We can isolate \(t_{apex}\) from this as:
\[t_{apex} = \frac{v_{0}}{g}\]Note that with these equations, at \(t = 0\) we got \(y = 0\), since we want to climb to some \(y_{apex}\) position we can write:
\[y_{apex} = v_{0}t_{apex}-\frac{1}{2}gt_{apex}^2\]We replace \(t_{apex}\) by \(v_{0}/g\):
\[\begin{aligned} y_{apex} &= v_{0}(\frac{v_{0}}{g})-\frac{1}{2}g(\frac{v_{0}}{g})^2 \\ y_{apex} &= \frac{v_{0}^2}{g}-\frac{v_{0}^2}{2g} \\ y_{apex} &= \frac{v_{0}^2}{2g} \\ % 2g(y_{apex}) &= v_{0}^2 \end{aligned}\]And now we have it: In order to reach some \(y_{apex}\) we need:
\[v_{0} = \sqrt{2g(y_{apex})}\]And it will take \(\sqrt{2g(y_{apex})} / g\) seconds to reach after jumping!
In the above example the blue line represents the \(y\) position over time, you can move the red line which represents the target \(y_{apex}\), while also adding/removing some velocity to/from \(v_{0}\) using the slider at the top.
As you can see: if your \(v_{0}\) is too high you will overshoot the altitude you meant to reach, while if you are short on speed then you’ll miss it!