The fastest velocity of the ball is approximately 5.95 m/s.
To solve this problem, we can use the conservation of mechanical energy. When the pendulum is released, the total mechanical energy is given by the potential energy at the starting point plus the kinetic energy at that point:
E = PE + KE
The potential energy of the pendulum at the starting point is given by its mass, the acceleration due to gravity, and its height above the starting point:
PE = mgh
The kinetic energy of the pendulum at the starting point is given by its mass, the velocity squared:
KE = 1/2 mv²
At the starting point, the velocity is 0, so the kinetic energy is also 0. Therefore, the total mechanical energy at the starting point is equal to the potential energy:
E = mgh
As the pendulum swings, the potential energy decreases and the kinetic energy increases. However, the total mechanical energy remains constant. When the pendulum reaches its lowest point, the velocity is at its maximum and the potential energy is at its minimum. Therefore, at the lowest point, the total mechanical energy is equal to the kinetic energy:
E = KE
Substituting the expressions for potential energy and kinetic energy, we get:
mgh = 1/2 mv²
Solving for v, we get:
v = sqrt(2gh)
Substituting the given values, we get:
v = sqrt(2 * 9.8 m/s² * 1.8 m)
= sqrt(35.28 m²/s²)
= 5.95 m/s
So the fastest velocity of the ball is approximately 5.95 m/s.