After the engines of a 100 kg spacecraft have been shut down, the spacecraft is found to be a d

After the engines of a 100 kg spacecraft have been shut...

After the engines of a $100 \mathrm{~kg}$ spacecraft have been shut down, the spacecraft is found to be a distance $10^7 \mathrm{~m}$ from the center of the earth, moving with a speed of $7000 \mathrm{~m} / \mathrm{s}$ at an angle of $45^{\circ}$ relative to a straight line from the earth to the spacecraft. (a) Calculate the total energy and angular momentum of the spacecraft. (b) Determine the semi-major axis and the eccentricity of the spacecraft's geocentric trajectory.

Key Concepts

Eccentricity

Eccentricity is a dimensionless parameter that characterizes the shape of an orbit. It quantifies how much an orbit deviates from being circular. An eccentricity of zero corresponds to a perfect circle, while values between zero and one indicate increasingly elliptical orbits. Eccentricity is determined using both the total energy and angular momentum of the orbiting body.

Semi-Major Axis

The semi-major axis is a key parameter of an ellipse in orbital mechanics, representing half the longest dimension of the orbital path. It not only describes the size of the orbit but also is directly related to the orbital period through Kepler's third law, making it a fundamental quantity in the description of bound orbital motion.

Angular Momentum

Angular momentum in orbital mechanics is a measure of the rotational inertia of the orbiting body with respect to a central point. It is calculated as the product of the mass, the radial distance from the central body, the velocity of the body, and the sine of the angle between the radial direction and the velocity vector. This conserved quantity is crucial for understanding the shape and orientation of the orbit.

Kinetic Energy

Kinetic energy in orbital dynamics is the energy associated with the motion of a body. It is calculated as half the product of the mass and the square of the velocity. In the context of celestial mechanics, it represents the energy due to the spacecraft's motion relative to the central body, such as the Earth.

Gravitational Potential Energy

Gravitational potential energy in orbital mechanics represents the energy due to the gravitational attraction between two bodies. It is typically defined as a negative quantity that depends on the masses of the bodies and their separation distance, reflecting the work required to separate the bodies to an infinite separation where the potential energy is zero.

Total Mechanical Energy

The total mechanical energy of an orbiting body is the sum of its kinetic and gravitational potential energies. This conserved quantity plays a central role in determining the nature of the orbit. Negative total mechanical energy indicates a bound orbit, while a zero or positive energy would correspond to parabolic or hyperbolic trajectories, respectively.

Transcript

00:01 So using work energy equation we can write u12 -d equals to delta t plus delta vg, this will be equals to 0.

00:08 And initial and final kinetic energies can be written as 1 by 2 m va square.

00:14 This is t1 and which will be equal to 1 by 2 multiplied by m multiplied by va is equal to 2 4 -3 -0 -3 -3 -0 multiplied by 1000 divided by 36 -0 because it is in the kilometer and whole square.

00:30 Here t1 comes out to be 2 .322 multiplied by 10 to the power 7 meter per second.

00:37 Okay.

00:37 And t2 will be equal to 1 by 2m vb square where vb is the speed at point b.

00:44 So now we make the center of the earth as datum.

00:48 So gravitation potential energies will be at point 1 will be vg1 equals to minus mgr square by ra.

00:58 Okay.

00:59 So substituting values we get my minus m multiplied by 9 .81 which is g and capital r is the radius of earth which is 6371 multiplied by 10 to the power 3 and bowl square divided by ra is equal to 7 ,000 km.

01:13 So 10 to the power 3.

01:15 So from here vg 1 is equal to minus 5 .6 double 8 multiplied by 10 to the power 7 meter.

01:23 Similarly vg 2 will be equals to minus m g r square by r b so from here we get minus m multiplied by 9 .81, multiplied by 6371 multiplied by 10 to the power 3, pole square, divided by rb is equals to 6500 km so 10 to the power 3 meter.

01:42 From here, we get minus 6 .1 to 6, 1 to 6 multiplied by 10 to the power 7 meter.

01:48 Okay, so now inserting all the values in this equation, we get 1 by 2m vb square minus 2 .32, multiplied by 10 to the power 7, minus 6 .126 multiplied by 10 to the power 7 plus 5 .6 .88 multiplied by 10 to the power 7...

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