In the following problems, except where otherwise indicated,...

In the following problems, except where otherwise indicated, assume that the radius of the earth is $6378 \mathrm{km}$, the earth's mass is $5.976\left(10^{24}\right) \mathrm{kg}$, the mass of the sun is $1.99\left(10^{30}\right) \mathrm{kg}, \quad$ and $\quad$ the $\quad$ gravitational $\quad$ constant $\quad$ is $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$. The rocket is initially in free-flight circular orbit around the earth. Determine the speed of the rocket at $A$. What change in the speed at $A$ is required so that it can move in an elliptical orbit to reach point $A^{\prime} ?$

In the following problems, except where otherwise indicated, assume that the radius of the earth is $6378 \mathrm{km}$, the earth's mass is $5.976\left(10^{24}\right) \mathrm{kg}$, the mass of the sun is $1.99\left(10^{30}\right) \mathrm{kg}, \quad$ and $\quad$ the $\quad$ gravitational $\quad$ constant $\quad$ is $G=66.73\left(10^{-12}\right) \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right)$.
The rocket is initially in free-flight circular orbit around the earth. Determine the speed of the rocket at $A$. What change in the speed at $A$ is required so that it can move in an elliptical orbit to reach point $A^{\prime} ?$

Key Concepts

Circular Orbital Motion

This concept involves understanding that an object in a circular orbit is maintained along its path by the gravitational force acting as the centripetal force. The equilibrium between gravitational pull and the need for centripetal acceleration establishes the orbital speed, which is a function of the mass of the central body and the radius of the orbit.

Elliptical Orbits and the Vis-Viva Equation

Unlike circular orbits, elliptical orbits have variable speeds at different points along the path. The vis-viva equation is a fundamental equation in orbital mechanics that relates the velocity of an orbiting object to its distance from the central mass and the semi-major axis of the ellipse. It is essential for determining the speed at any point in an elliptical orbit, such as during orbital transfers.

Orbital Maneuvers and Delta-v

Orbital maneuvers refer to the intentional changes in an object's velocity, known as delta-v, to achieve a desired orbit change. These maneuvers are often executed impulsively at specific points in an orbit (for example, points where the orbit touches another desired orbit) and are calculated by considering the differences in energy and velocity between the current and target orbits.

Conservation of Mechanical Energy in Orbital Mechanics

This fundamental principle states that in an isolated system, where only conservative forces (like gravity) are acting, the total mechanical energy remains constant. In the context of orbital mechanics, the conservation of energy is used to relate the kinetic and potential energies at different points along an orbit, facilitating the calculation of changes in velocity required for orbital transfers.

Transcript

00:01 Hi everyone, so what we have here is a rocket that's in initially a circular orbit around the earth.

00:07 And what we're asked to do is solve for first the speed at a, as shown here.

00:14 And we're also required to solve for the change in speed or the delta v at a that's required so we can go from that circular orbit into the elliptical orbit that's shown here.

00:27 And i actually need to change this.

00:30 So this is the problem statement that we have here.

00:35 We see that the radius of the circular orbit is 8 megameters, and the distance from a prime to the surface of the earth is 19 megameters.

00:47 And the concepts that we're going to use to solve this problem is our knowledge of orbital dynamics, and also orbital mechanics as well.

00:58 So let's go ahead and solve this problem.

01:00 So what we need to do is first determine the speed that's required to stay at the circular orbit containing a.

01:08 So we know that r0, which is going to be a radius, is going to be equal to the radius of the earth, plus the distance from the surface of the earth to point a.

01:18 We're just going to call that r .e.

01:20 And plugging in the quantities that we know, we know that r0 is equal to 14 .38 times 10 to the 6 meters.

01:29 We can directly solve for our velocity at c or velocity at a, and that's we're going to denote as vac for circular, and that's going to be equal to the square root of g, m, e, all over r0...

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