To solve these problems, we can use the laws of motion and the principles of conservation of energy.
1) To find the initial speed of the rock, we can use the principle of conservation of energy, which states that energy is conserved in a system. At the top of the cliff, the total energy of the rock is given by its potential energy, which is m * g * h, where m is the mass of the rock, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the cliff. At the base of the cliff, the total energy of the rock is given by its kinetic energy, which is (1/2) * m * v^2, where v is the speed of the rock.
Since energy is conserved, we can equate the total energy at the top of the cliff to the total energy at the base of the cliff:
m * g * h = (1/2) * m * v^2
By rearranging the equation, we can solve for the initial speed of the rock:
v = sqrt(2 * g * h)
Plugging in the values, we get:
v = sqrt(2 * 9.8 m/s^2 * 32m) = 28.1 m/s
Therefore, the initial speed of the rock is approximately 28.1 m/s.
2) To find the greatest height of the rock as measured from the base of the cliff, we can use the fact that the potential energy of the rock is converted to kinetic energy at the base of the cliff. Thus, at the top of the rock's trajectory, its potential energy is zero, and at the base of the cliff, its kinetic energy is zero.
Using the principle of conservation of energy, we can equate the potential energy at the top of the trajectory to the kinetic energy at the base of the cliff:
m * g * h = (1/2) * m * v^2
Solving for h, we get:
h = (1/2) * v^2 / g
Plugging in the values, we get:
h = (1/2) * (29 m/s)^2 / 9.8 m/s^2 = 44.8 m
Therefore, the greatest height of the rock as measured from the base of the cliff is approximately 44.8 m.