Question

Определить работу которую необходимо затратить чтобы вывести ракету за пределы поля тяготения Земли, если ракета стартует с космического корабля, движущегося по круговой орбите на уровне 500 км над поверхностью Земли масса ракеты 200 кг

Answer 1

 Работа будет равна разности кинетических энергий W2-W1=mV2^2/2 - mV1^2/2, где V2 = второй космической скорости, а V1- это скорость на орбите.

Answer 2

Calculation of Work Required to Launch a Rocket Beyond Earth's Gravitational Field

To determine the work required to launch a rocket beyond the gravitational field of Earth, we need to consider the initial kinetic energy of the rocket and the work done against gravity.

The initial kinetic energy of the rocket can be calculated using the formula:

Kinetic Energy = (1/2) * mass * velocity^2

Given that the mass of the rocket is 200 kg and it is launched from a spacecraft moving in a circular orbit at an altitude of 500 km above the Earth's surface, we need to calculate the velocity of the rocket.

To calculate the velocity, we can use the formula for the velocity of an object in circular motion:

Velocity = sqrt(G * M / R)

Where: - G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2) - M is the mass of the Earth (approximately 5.972 × 10^24 kg) - R is the radius of the orbit (radius of the Earth + altitude of the orbit)

Let's calculate the velocity:

R = radius of the Earth + altitude of the orbit

The radius of the Earth is approximately 6,371 km. Converting the altitude of the orbit from km to meters, we get:

R = 6,371 km + 500 km = 6,871 km = 6,871,000 meters

Now we can calculate the velocity:

Velocity = sqrt(G * M / R)

Calculating the velocity using the given values and the formula, we find:

Velocity ≈ 7,850 m/s Now that we have the velocity, we can calculate the initial kinetic energy of the rocket:

Kinetic Energy = (1/2) * mass * velocity^2

Substituting the values, we get:

Kinetic Energy = (1/2) * 200 kg * (7,850 m/s)^2

Calculating the kinetic energy, we find:

Kinetic Energy ≈ 15,470,000 J Next, we need to calculate the work done against gravity. The work done against gravity is equal to the change in potential energy of the rocket as it moves from the initial altitude to an infinite distance from Earth.

The potential energy of an object near the surface of the Earth is given by the formula:

Potential Energy = mass * g * height

Where: - mass is the mass of the rocket (200 kg) - g is the acceleration due to gravity (approximately 9.8 m/s^2) - height is the altitude of the orbit (500 km)

Converting the altitude of the orbit from km to meters, we get:

height = 500 km = 500,000 meters

Calculating the potential energy using the given values and the formula, we find:

Potential Energy = 200 kg * 9.8 m/s^2 * 500,000 meters

Calculating the potential energy, we find:

Potential Energy ≈ 9,800,000 J The work done against gravity is equal to the negative change in potential energy. Therefore, the work done against gravity is:

Work = -Potential Energy

Substituting the value of potential energy, we get:

Work = -9,800,000 J

Finally, to determine the total work required to launch the rocket beyond Earth's gravitational field, we need to sum the initial kinetic energy and the work done against gravity:

Total Work = Kinetic Energy + Work

Substituting the values, we get:

Total Work ≈ 15,470,000 J + (-9,800,000 J)

Calculating the total work, we find:

Total Work ≈ 5,670,000 J

Therefore, the work required to launch the rocket beyond the gravitational field of Earth is approximately 5,670,000 joules.