How would we go about deriving an equation which would tell us how far the maximum distance a child would travel if they jumped of a swing?
Should I start with the two differential equations for maximum range for a projectile and also a shm of a pendulum and then combine them?
You will need the initial height, velocity of the child. Pretty easy, if you know the max height or max velocty at bottom of swing, the KE at any point + PE at any point is equal to the max energy of the system.
Now, knowing the position (initial height, initial velocity, and angle (get the angle of the tangent to the circle arc at release), you now can use you kinematic equation for a body under gravitational acceleration.