Question

calculate the time required for a spacecraft launched into a paraboliv trajectory at a perigee altitude of 200 km to leave the earth's sphere of influence

Answer 1

The time required for a spacecraft to leave Earth's sphere of influence would depend on its engine capabilities and trajectory, but it must reach at least the escape speed of 11 kilometers per second. Without detailed information on the spacecraft's initial speed and acceleration, an exact time cannot be calculated.

Step-by-step explanation:

To estimate the time required for a spacecraft launched into a parabolic trajectory at a perigee altitude of 200 km to leave Earth's sphere of influence, we need to understand that the spacecraft must reach the escape speed. The escape speed from Earth is approximately 11 kilometres per second. Once the spacecraft achieves this speed, it is no longer bound by Earth's gravity and will continue to travel into space, eventually leaving Earth's sphere of influence.

As we don't have the specific initial velocity and acceleration details of the spacecraft, we cannot use the formula v = vo + at to calculate the exact time. However, we know that the escape velocity is the minimum speed needed for an object to 'break free' from the gravitational attraction of a massive body, without further propulsion. This speed does not depend on the direction of travel.

Without additional data provided in the question, we can only state that a spacecraft would have to maintain or exceed this speed of 11 kilometres per second to leave Earth's sphere of influence. The time required to reach this speed would depend on the characteristics of the spacecraft's engines and its trajectory. Once at escape velocity, the spacecraft will continue moving away indefinitely, the time associated with actually exiting the sphere of influence after reaching escape speed is not clearly defined as it is an asymptotic escape.

Answer 2

To calculate the time required for a spacecraft to leave the Earth's sphere of influence, we can use Kepler's third law and the given altitude of the spacecraft's trajectory. By substituting the values into the formula, we can find the period of the orbit and determine the time required.

Step-by-step explanation:

To calculate the time required for a spacecraft to leave the Earth's sphere of influence, we need to determine the escape speed from Earth's surface and the altitude of the spacecraft's trajectory. The escape speed from Earth is approximately 11 kilometers per second. Assuming the spacecraft follows a parabolic trajectory with a perigee altitude of 200 km, we can calculate the time required by using Kepler's third law.

The time for one orbit of an artificial satellite is related to the radius of the orbit by Kepler's third law. The formula is expressed as T^2 = (4π²/GM) * r^3, where T is the period of the orbit, G is the gravitational constant (approximately 6.67 × 10^-11 Nm²/kg²), M is the mass of Earth (approximately 5.97 × 10^24 kg), and r is the distance between the center of Earth and the satellite's orbit.

Substituting the values given, we have r = (6370 km + 200 km) = 6570 km = 6570000 m. Solving for T, we get T = √((4π²/GM) * r^3). Plugging in the values, we find T ≈ 1.67 hours.