What is the minimum velocity in which the cue ball must be hit in order to hit the blue ball into the hole? Assume straight line (cue ball stops moving after hitting the blue ball).

Distance between cue ball and blue ball = 1.2 meters.
Distance between blue ball and hole = 0.75 meters .
Coefficient of friction between ball and table = 0.005.

work on the cue ball: M*g*mu*1.2 m

and the blue ball M*g*mu*.75m

those added must equal the initial KEnergy on the cue: 1/2 M V^2
so solve for V. Notice M divides out;

To determine the minimum velocity at which the cue ball must be hit in order to hit the blue ball into the hole, we can use the principle of conservation of mechanical energy.

The principle of conservation of energy states that the total mechanical energy of a system is constant if there are no external forces acting on it. In this case, we can assume that there are no external forces acting on the system consisting of the cue ball, the blue ball, and the table.

The initial mechanical energy of the system is given by the kinetic energy of the cue ball and the potential energy of the blue ball. The final mechanical energy of the system is given by the potential energy of the blue ball when it reaches the hole.

Let's break down the steps to calculate the minimum velocity of the cue ball:

Step 1: Calculate the initial mechanical energy of the system.
The kinetic energy of the cue ball is given by the equation: KE = (1/2) * m * v^2, where m is the mass of the cue ball and v is its velocity. Since we are only interested in the minimum velocity, we can assume the mass of the cue ball cancels out when comparing initial and final kinetic energies.

Step 2: Calculate the potential energy of the blue ball when it is at the hole.
The potential energy is given by the equation: PE = m * g * h, where m is the mass of the blue ball, g is the acceleration due to gravity, and h is the height of the blue ball above the hole.

Step 3: Equate the initial and final mechanical energies to find the minimum velocity.
KE (cue ball) = PE (blue ball at hole)

Substituting the equations from Step 1 and Step 2 into Step 3, we can solve for v:

(1/2) * m * v^2 = m * g * h

Simplifying the equation by canceling out the mass m, we get:

(1/2) * v^2 = g * h

Solving for v, we get:

v = √(2 * g * h)

Now, let's substitute the given values into the equation. The acceleration due to gravity (g) is approximately 9.8 m/s^2, and the height of the blue ball above the hole (h) is 0.75 meters.

v = √(2 * 9.8 * 0.75) = √14.7 ≈ 3.83 m/s

Therefore, the minimum velocity at which the cue ball must be hit in order to hit the blue ball into the hole is approximately 3.83 m/s.