Questions from a jumping kangaroo

Question

Comments

You can find out if your equations are correct by applying them to a particular case with a known solution which you can find on the internet - eg cylinder, half-cylinder, cuboid, triangular prism.

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That is a good suggestion but I dont know how to search for this type of problem.

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You could try.

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I did try thats why I posted. The only one I found was https://www.physicsforums.com/threads/minimum-velocity-required-by-grasshopper.886390/ which is a start.

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Good find. Also see  https://math.stackexchange.com/questions/2317985/firing-a-projectile-over-a-circular-obstacle and https://www.usna.edu/Users/physics/mungan/_files/documents/Scholarship/JumpingFrog.pdf

Answer 1

A general algorithm which works for all functions $f(x)$ will have to use a numerical method of solution - ie start with an initial approximation to the trajectory, estimate how close this comes to the optimal solution (the "error"), use this to make an adjustment to the solution, estimate the new "error," and continue the same cycle until the "error" is small enough for your purposes.

Analytic solutions in the form of closed functions are possible only for particular shapes of $f(x)$ such as rectangles, circles, and ellipses.

If the object is convex and has a vertical plane of symmetry then you can use the method in my answer to Minimum speed required to clear rectangular object. This involves finding points A, B on the same horizontal level, at which the projectile could graze the obstacle at an angle of $45^{\circ}$ to the horizontal. The distance AB enables you to find the required speed at these points, and the height of AB above the ground enables you to find the minimum speed with which the object should be launched.

This appears to be the method which you are using in your calculation.