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Answer:

1. A rocket of mass 2000 kg is launched vertically upwards from the ground with an initial velocity of 200 m/s.

a. The total mechanical energy of the rocket can be calculated using the formula:

Total mechanical energy = kinetic energy + potential energy

At launch, the rocket has only kinetic energy. The kinetic energy of the rocket can be calculated as:

KE = 0.5 * mass * velocity^2

where mass = 2000 kg and velocity = 200 m/s

KE = 0.5 * 2000 kg * (200 m/s)^2

KE = 40,000,000 J

Therefore, the total mechanical energy of the rocket at launch is 40,000,000 J.

b. At the maximum height, the velocity of the rocket will be zero. The potential energy of the rocket at this height is equal to the total mechanical energy at launch. Therefore:

Total mechanical energy = potential energy

Potential energy = mgh

where m = 2000 kg, g = 9.81 m/s^2, and h is the maximum height reached by the rocket.

40,000,000 J = 2000 kg * 9.81 m/s^2 * h

h = 2040.82 m

Therefore, the maximum height reached by the rocket is approximately 2040.82 m.

To check the answer using kinematic equations, we can use the equation:

v^2 = u^2 + 2gh

where v = 0 m/s (velocity at maximum height), u = 200 m/s (initial velocity), g = 9.81 m/s^2 (acceleration due to gravity), and h is the maximum height.

0^2 = (200 m/s)^2 + 2 * 9.81 m/s^2 * h

h = 2040.82 m

c. To find the velocity of the rocket at h = 500 m, we can use the conservation of energy principle:

Total mechanical energy = kinetic energy + potential energy

At any height h, the total mechanical energy is equal to the kinetic energy and potential energy of the rocket at that height. Therefore:

Total mechanical energy = KE + PE

where KE is the kinetic energy and PE is the potential energy.

At h = 500 m, the potential energy of the rocket is:

PE = mgh

where m = 2000 kg, g = 9.81 m/s^2, and h = 500 m.

PE = 2000 kg * 9.81 m/s^2 * 500 m

PE = 9,810,000 J

The total mechanical energy at h = 500 m is therefore:

Total mechanical energy = KE + PE

Total mechanical energy = 40,000,000 J - 9,810,000 J

Total mechanical energy = 30,190,000 J

At any height h, the kinetic energy of the rocket is given by:

KE = 0.5 * m * v^2

where m is the mass of the rocket and v is its velocity.

At h = 500 m, the kinetic energy of the rocket is therefore:

KE = 0.5 * m * v^2

We can solve for v:

v = sqrt(2 * KE / m)

v = sqrt(2 * 30,190,000 J / 2000 kg)

v = 194.79 m/s

Therefore, the velocity of the rocket at h = 500 m is approximately 194.79 m/s.

2. A 10 kg boulder is dropped from a 10 m high cliff. The gravitational

acceleration at this location is 10 N/kg.

a. At a height of 10 meters above the ground, the boulder has only potential energy given by:

PE = mgh = (10 kg)(10 m/s²)(10 m) = 1000 J

So the mechanical energy of the boulder at this height is 1000 J.

b. Using conservation of energy, we can equate the mechanical energy of the boulder at the initial height to its mechanical energy at a height of 8 meters above the ground, where it has both potential and kinetic energy:

PE + KE = mgh + (1/2)mv²

We know that the mass of the boulder is 10 kg, the initial height is 10 m, and the gravitational acceleration is 10 N/kg. At a height of 8 meters above the ground, the potential energy is:

PE = mgh = (10 kg)(10 m/s²)(8 m) = 800 J

Thus, the kinetic energy is:

KE = ME - PE = 1000 J - 800 J = 200 J

Using the formula for kinetic energy, we can find the velocity of the boulder at this height:

KE = (1/2)mv²

v² = (2KE) / m = (2)(200 J) / 10 kg = 40 m²/s²

v = sqrt(40 m²/s²) ≈ 6.3 m/s

So the velocity of the boulder, when it is at a height of 8 meters above the ground, is approximately 6.3 m/s.

c. At any height h, the mechanical energy of the boulder is given by:

ME = mgh + (1/2)mv²

We want to find the height at which the velocity of the boulder is 11.8 m/s. We can set up the following equation:

ME = mgh + (1/2)mv² = constant

We know that the mass of the boulder is 10 kg and the mechanical energy is constant throughout the motion, so we can simplify the equation:

gh + (1/2)v² = constant/m

Substituting the given values, we get:

(10 m/s²)h + (1/2)(11.8 m/s)² = constant/10 kg

Solving for h, we get:

h = (constant/10 kg - (1/2)(11.8 m/s)²) / (10 m/s²)

We can use the initial height of the boulder (10 m) and its initial velocity (0 m/s) to find the value of the constant:

constant = mgh + (1/2)mv² = (10 kg)(10 m/s²)(10 m) + (1/2)(10 kg)(0 m/s)² = 500 J

Substituting this value, we get:

h = (500 J/10 kg - (1/2)(11.8 m/s)²) / (10 m/s²)

h ≈ 28.8 m

So the height of the boulder, when it is traveling at a velocity of 11.8 m/s, is approximately 28.8 meters.

d. At the ground level, the boulder has zero potential energy, so all its mechanical energy is kinetic energy. Using conservation of energy, we can equate the initial mechanical energy of the boulder to its final mechanical energy at the ground level:

ME = mgh + (1/2)mv² = (1/2)mvf²

We know that the mass of the boulder is 10 kg, the initial height is 10 m, and the initial velocity is