# Can a sphere simultaneously contact two planes perpendicular to each other at any angle other than 45 degrees?

Question arises from the game of racquetball where the ball strikes the wall and floor simultaneously with scoring implications depending on the outcome.

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What exactly is supposed to be at the "angle other than 45 degrees? Are you talking about the path of a moving sphere before it hits the planes? – Robert Israel Aug 15 '12 at 1:43

It's always the same angle, but its not $45^\circ$. To see why, think of the plane perpendicular to the wall and the floor that passes through the center of the sphere. This gives a 'slice' of the sphere, and includes the points where the sphere touches the wall and the floor.

Effectively, this brings it down a dimension, and it's much easier to see that a circle 'wedged in a right angle' will always look that same. That is, the lines will be tangent to the circle (right angles to the radius) and the two tangents will be separated by a quarter-circle, or $90^\circ$.

Here is a picture with two circles showing the tangents:

EDIT

In a comment, Gerry mentions that you might be interested in the path of the ball before it hits both the wall and floor at the same time. That wasn't how I interpreted the question, but I can address it too.

Let us again look at a 2D cross section of the world, and suppose that the wall is the $y$ axis, the floor is the $x$ axis, the ball is radius $1$ and the room is to the left (so that it agrees with the picture above). Then if the ball's center is as $(-1,1)$, then the ball will be touching both the floor and wall at the same time.

A reasonable assumption is that the ball falls in a parabolic path (a common assumption in elementary physics). Then any downward parabola with the right leg passing through $(-1,1)$ will describe a possible flight path of a ball that touches both the floor and wall at the same time. This describes a whole range of incident angles, so it's not true that the path of the ball will always come off the floor/wall at $45^\circ$.

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What you have drawn is certainly the configuration at the instant when the ball hits wall and floor. OP might be asking a different question: in what direction was the ball traveling in the instant before it hit wall and floor? Clarification from OP is necessary. – Gerry Myerson Aug 15 '12 at 1:50
@Gerry: Oh! That makes so much sense, as in where the $45^\circ$ would come from! – mixedmath Aug 15 '12 at 2:26
Good edit. Indeed, the ball could be rolling along the floor, and come at the corner at an angle of zero. – Gerry Myerson Aug 15 '12 at 4:56

Your real question is not clear, but concerning a ball bouncing off a wall and floor, there seems nothing mysterious. A ball can approach the line where the wall touchs the floor from any angle (as long as it is on the correct side of the wall and of the floor). Doing so, it is clearly always possible that it hits the floor first, and it is also possible that it hits the wall first. Between those two trajectories there must be one where it hits floor and wall simultaneously, and this has no reprecussions at all on the direction the ball was coming from. I could add that assuming perfectly elastic collisions, the ball will return in (almost) the same direction it came from, regardless of which surface was hit first. So what did you really want to know about the situation?

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