A blue billiard ball with mass m crashes into a red billiard ball with the same mass that is at rest. The collision results in the blue billiard ball travelling with a velocity of v and the red billiard ball with a velocity of 3V. In terms of v and m determine the blue billiard ball's velocity before the collision.

Let the man run to point P, which is x meters upstream from the line AB. He swims across, being swept y meters downstream, and then runs to B.

We know that he is in the water for d/(v/3) = 3d/v seconds, so
y = 3d/v * v = 3d.

His path is thus

√(d^2+x^2) meters on land,
d meters in the water,
√((2d)^2+(x-3d)^2) meters on land to B

Now, dividing distance by speed, his time to travel is

t = 1/(3v) (√(d^2+x^2)+√((2d)^2+(x-3d)^2)) + 3d/v
=
√(d^2+x^2)+√(x^2-6dx+13d^2)+9d
--------------------------------------
           3v

Let the man run to point P, which is x meters upstream from the line AB. He swims across, being swept y meters downstream, and then runs to B.

We know that he is in the water for d/(v/3) = 3d/v seconds, so
y = 3d/v * v = 3d.

His path is thus

√(d^2+x^2) meters on land,
d meters in the water,
√((2d)^2+(x-3d)^2) meters on land to B

Now, dividing distance by speed, his time to travel is

t = 1/(3v) (√(d^2+x^2)+√((2d)^2+(x-3d)^2)) + 3d/v
=
√(d^2+x^2)+√(x^2-6dx+13d^2)+9d
---------------------------------------------
                  3v

To determine the blue billiard ball's velocity before the collision, we can apply the law of conservation of momentum. According to this law, the total momentum before the collision is equal to the total momentum after the collision.

The momentum of an object is calculated by multiplying its mass by its velocity. So, we can write the equation for momentum conservation as follows:

(mass of blue ball * velocity of blue ball) + (mass of red ball * velocity of red ball) = (mass of blue ball * final velocity of blue ball) + (mass of red ball * final velocity of red ball)

Let's denote the mass of the blue ball as m, the velocity of the blue ball before the collision as v1, and the velocity of the red ball before the collision as v2 (which is zero since it is at rest). The final velocity of the blue ball is v, and the final velocity of the red ball is 3V.

Using these variables, the momentum conservation equation becomes:

(m * v1) + (m * 0) = (m * v) + (m * 3V)

Since the mass of the red and blue balls is the same, we can simplify the equation further:

m * v1 = m * v + 3m * V

Now, we can solve for v1 to find the blue ball's velocity before the collision:

m * v1 = m * v + 3m * V
v1 = v + 3V

Therefore, the blue billiard ball's velocity before the collision is v + 3V.