At engine burnout on a mission, a shuttle had reached point...

At engine burnout on a mission, a shuttle had reached point $A$ at an altitude of $40 \mathrm{mi}$ above the surface of the earth and had a horizontal velocity $\mathbf{v}_{0} .$ Knowing that its first orbit was elliptic and that the shuttle was transferred to a circular orbit as it passed through point $B$ at an altitude of $170 \mathrm{mi}$, determine $(a)$ the time needed for the shuttle to travel from $A$ to $B$ on its original elliptic orbit, $(b)$ the periodic time of the shuttle on its final circular orbit.

Key Concepts

Kepler's Third Law

Kepler's third law establishes a fundamental relationship between the orbital period and the size of the orbit, stating that the square of the period is proportional to the cube of the semi-major axis. This law is essential for determining the time it takes for an object to complete an orbit, whether elliptical or circular. It provides a straightforward method for calculating orbital periods once the dimensions of the orbit are known.

Orbital Mechanics Fundamentals

This concept involves the principles governing motion under gravity, including conservation of energy and angular momentum. It lays the basis for analyzing satellite trajectories, describing how the balance between gravitational attraction and the inertia from a satellite’s velocity determines its orbit. Understanding these fundamentals is crucial for determining orbital parameters and predicting the motion along different types of orbits.

Elliptical Orbit

An elliptical orbit is a closed path where the distance between the orbiting object and the central body varies, characterized by semi-major and semi-minor axes. The position of the object along the ellipse is often described by the true anomaly, and its motion is governed by Kepler’s laws. When calculating the time of flight between two points along an ellipse, one typically uses Kepler’s equation and the geometry of the ellipse to relate the change in anomaly to elapsed time.

Circular Orbit

A circular orbit is a special case of an elliptical orbit where the distance between the orbiting object and the central body remains constant. In such orbits, all points are equidistant from the center of mass, and the orbital speed is uniform. The period of a circular orbit can be calculated directly using formulas derived from Newton’s law of gravitation and centripetal force requirements, often encapsulated in Kepler’s third law.

Transcript

00:01 In this problem, we have a space shuttle in a circular orbit, 185 miles above your surface, so here.

00:12 And it launches an inertial upper stage carrying communication satellites.

00:20 We place in geosecuit orbit, which we know is at 22 ,2 ,230 miles.

00:29 And so it launches it here and it puts it into an elliptic orbit that will then get it out to this point.

00:35 And then we need to then get it changed the velocity so that we get into a circular orbit here.

00:42 So we want to determine the velocity of the ius relative to the shuttle after its engine has been fired at a and the increase of velocity required at b, the place the satellite in its final orbit.

00:53 So we need to shoot it off here, give it some extra velocity, so get it in this orbit.

00:59 And then change the velocity again to get into this orbit.

01:04 So we have our initial potential energy is, i guess i've used m1 as the mass of the shuttle and m2 is the mass of the launch vehicle.

01:18 And so the potential energy at this point here, and then the kinetic energy, again because they're both in a circular orbit.

01:26 So because the, this is the space space bottles in a circular orbit, we can write it this way.

01:35 And then we have some kinetic energy of the launch vehicle in the circumferential direction.

01:47 So we, this is after it's been launched.

01:51 And then we know that the potential energy in the final state when it's out here, we have the potential energy of the space shuttle and the potential energy of the launch vehicle out here...

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