FinMaMan's Math Problem of the Week #1

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Hello again lovely peoples.

I like to pose math problems to everyone I meet for the first time. Being new to this forum, here goes.

A pool table is 4 meters long and a meter wide. There are three pool balls on a pool table, each standing at 2 meters from the front of the table, and one standing directly in the center, one directly 25 cm next to it, and another 25 cm in the other. The cue ball has a radius of 2 cm. What is the probability of hitting another ball if one shoots at a completely random angle that is between 0 and 180 degrees (0 or pi radians, if you fellow mathmos prefer ) inclusively?

Later this week, I shall divulge the answer

And (info that I forgot to add), the ball lies 10 cm away from the rim of the table.

Is this on a frictionless table?

Eric, I forgot to tell you: all shots are given at equal strength, and they have a 100% chance of reaching the same level of the balls if they don't bounce off the sides.

And are the object balls (none cue balls) also 2 cm in radius? If they were as fat as bowling balls it might make a difference.

Needless to say the balls are oddly small- and the table is a really strange size and shape for billiards.

You dont mention a position for the cue ball.

Can you place the cue ball anywhere on the table?

Or just from one of the short sides (head or foot) of the table?

Last edited by naturalplastic on 24 Jun 2013, 8:30 pm, edited 1 time in total.

I'm trying to picture this in my mind.

If the table is 4 meters long and three balls are two meters from the front, then they would be on a line across the middle of a table. Each ball is 4 cm in diameter for a total of 12 cm so they would block 12% of the meter wide table. (Naturalplastic is right -- that is an oddly shaped pool table.)

Furthermore, the center of the cue ball only needs to come within 2 cm of the edge of one of the other balls. Note that it is wider than the distance between the center ball and the ones to either side. Thus, there would be 16% of the width of the table where its trajectory would encounter another ball.

If the table were a billiards table instead of a pool table, i.e. no pockets, and it was frictionless, then once put into motion the cue ball would stay in motion and every possible path should eventually hit one of the three balls. Even if shot parallel to two cusions such that the ball were to just bounce back and forth between two cushions, there would be small perturbations from the coriolis effect that would eventually add up.

But it's not frictionless and the cue ball is hit hard enough to reach the middle of the table if it doesn't contact the sides.

So it looks like we would have to consider every possible position for the cue ball on one end of the table and every possible angle shot that is not away from the balls. (I assume that the 0 to pi radian angle is such that it doesn't move further from the balls in the lengthwise direction of the table).

Also note that the arrangement of balls covers 17% of width of the table only along the center line.

So am I viewing this correctly?

Just out of curiousity, why not permit shots in any direction instead of just in half half the possible directions?

And if it is in just half the possible directions how is that 0 to pi radians oriented? The problem, as stated, gives no clue about that.

I'm also curious if there is any particular spot for the cue ball or can it be anywhere in the table? For example, can the cue ball be leaning against one or two of the object balls?

This math problem requires some specificity about the physics involved to give a probability. In particular, the coefficients of static and kinetic friction of the table surface, the elasticity of collisions between balls or between a ball and the bumper (edge of the table), the initial velocity of the cue ball when struck, the probability that a ball, cue ball or one of the three initial balls, will bounce off the table under conditions to be specified by the poser of this math challenge. Other initial conditions come to mind.

Without these initial conditions, I can say that under a variety of circumstances the probability is 1, and under other circumstances the probability is 0. Without all of the information above I cannot give specific probabilities.

Any action on this lately?

I suspect that the OP concluded that the question just can't fly no matter how much work is put into it. So he abandoned it.

Not even trying..I think a headache is knocking on my door