the coefficient of restitution between the ball and the floor is 0.60 If the ball is drop from rest at the height of 6.6 m from the floor. Find a) what is the maximum height will the ball attain after the first bounce. b) how much kinetic energy is lost duirng the impact if the mass of the ball is 0.9 kg.

a) h= .6*6.6

b) Ke lost= original KE-final KE=
= mg(6.6)(1-.6)

e^2 = h2/h1

h2 = e^2(h1)
h2 = .6(6.6 m)
h2 = 3.96 m

To find the maximum height the ball will attain after the first bounce, we need to consider the conservation of mechanical energy.

a) The equation for conservation of mechanical energy is:
Initial potential energy = Final potential energy + Final kinetic energy

Initial potential energy = m * g * h
Final potential energy = m * g * h'
Final kinetic energy = 0.5 * m * v^2

Where:
m = mass of the ball (0.9 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = initial height (6.6 m)
h' = maximum height after the first bounce
v = velocity just before impact

Using the equation for conservation of mechanical energy, we can write:
m * g * h = m * g * h' + 0.5 * m * v^2

Since the ball is dropped from rest, the initial velocity is 0, so the equation becomes:
m * g * h = m * g * h' + 0.5 * m * 0^2

Simplifying the equation:
m * g * h = m * g * h'
m * g * h - m * g * h' = 0
m * g * (h - h') = 0
h - h' = 0

Therefore, the maximum height the ball will attain after the first bounce is 0.

b) The coefficient of restitution (e) is defined as the ratio of the final relative velocity to the initial relative velocity between two objects after their collision.

The equation to calculate the loss of kinetic energy during impact using the coefficient of restitution is:
Loss of kinetic energy = (1 - e^2) * Initial kinetic energy

Given:
Coefficient of restitution (e) = 0.60
Mass of the ball (m) = 0.9 kg
Initial velocity (v) = ?

To find the initial velocity, we can use the equation for potential energy:
Potential energy = m * g * h

m * g * h = 0.5 * m * v^2

Simplifying the equation:
g * h = 0.5 * v^2

Substituting the known values:
9.8 * 6.6 = 0.5 * v^2

v^2 = (9.8 * 6.6) / 0.5
v^2 = 128.7
v ≈ 11.35 m/s

Now, we can calculate the loss of kinetic energy:
Initial kinetic energy = 0.5 * m * v^2

Loss of kinetic energy = (1 - e^2) * Initial kinetic energy
Loss of kinetic energy = (1 - 0.60^2) * (0.5 * 0.9 * (11.35)^2)

Calculating the value:
Loss of kinetic energy ≈ 0.0242 * 57.7
Loss of kinetic energy ≈ 1.40 J

Therefore, the loss of kinetic energy during the impact is approximately 1.40 Joules.

To find the maximum height the ball will attain after the first bounce, we need to consider the conservation of energy.

a) First, we need to determine the initial potential energy of the ball when it is dropped from a height of 6.6 m. The potential energy formula is given by:

Potential Energy (PE) = mass (m) * gravity (g) * height (h)

Given:
Mass of the ball (m) = 0.9 kg
Height (h) = 6.6 m
Acceleration due to gravity (g) = 9.8 m/s²

PE = 0.9 kg * 9.8 m/s² * 6.6 m = 56.364 J (Joules)

Next, we need to consider the loss of energy during the impact with the floor. The coefficient of restitution (e) is a measure of how much kinetic energy is retained after a collision. In this case, e = 0.60, which means 60% of the kinetic energy is retained after the impact.

Let's assume that the ball reaches its maximum height after rebounding.

The total initial energy, including potential energy and kinetic energy, is given by:

Total Initial Energy = PE + KE

Since the ball is dropped from rest, the initial kinetic energy is zero, so:

Total Initial Energy = PE + 0 = PE

Now, we know that the ball retains 60% of its kinetic energy after the impact, so the final kinetic energy is given by:

Final KE = e * Initial KE

Since the final kinetic energy is equal to the kinetic energy at the maximum height, we can rewrite the equation as:

Final KE = e * Total Initial Energy

Substituting the given values:

Final KE = 0.60 * 56.364 J = 33.8184 J (Joules)

To find the maximum height, we equate the final kinetic energy with the potential energy at that height:

Final KE = PE at maximum height

33.8184 J = m * g * h

Substituting the known values:

33.8184 J = 0.9 kg * 9.8 m/s² * h

h = 33.8184 J / (0.9 kg * 9.8 m/s²)

Therefore, the maximum height the ball will attain after the first bounce is:

b) Using the formula for kinetic energy (KE), which is:

KE = 0.5 * m * v²

We can calculate the initial kinetic energy (KE_initial) when the ball is dropped from rest:

KE_initial = 0.5 * m * v_initial²

Since the ball is dropped from rest, its initial velocity (v_initial) is zero. Hence, the initial kinetic energy is also zero:

KE_initial = 0.5 * 0.9 kg * 0² = 0 J (Joules)

The kinetic energy lost during the impact is the difference between the initial and final kinetic energy:

Kinetic Energy Lost = KE_initial - Final KE

Substituting the known values:

Kinetic Energy Lost = 0 J - 33.8184 J = -33.8184 J

The negative sign indicates that energy is lost during the impact.

Therefore, the kinetic energy lost during the impact is 33.8184 J (Joules).