Kinematics With Portals

Classical Mechanics Level 4

If you have never played the video game Portal, a portal is essentially a wormhole (or tunnel) which can teleport an object from one location to another instantaneously, almost as if the object had simply passed through a door. The portal will conserve kinetic energy and speed. However, the object's direction of travel will get shifted depending on how the portals are oriented relative to each other. In math-speak, the object's velocity vector relative to the first portal is equal to that same object's velocity vector relative to the second portal.

Now for the problem:

A ball is launched horizontally off of a ledge of height \(H\) at an initial speed of \(V_o\). Portal A (orange) is placed on the ground at a distance \(R\) such that the ball will pass directly through the portal's center.

On a wall a distance \(\dfrac{R}{2}\) away, Portal B (blue) is placed at a height \(h\) such that, after the ball passes through, it will again fall into Portal A, but from the other side.

Note that the angle at which the ball falls into Portal A (relative to the ground) is the same as the angle at which the ball falls out of Portal B (relative to the wall).

Given that \(H = 6.00 \text{ m}\) and \(V_o = 2.53\text{ m/s}\), find \(h\).

Give your answer in meters, rounded to 3 significant figures.

Take the acceleration due to gravity as \(g = 9.81 \text{ m/s}^2 \).


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