A 150.0 kg rocket moving radially outward from Earth has a speed of 3.70 km / s when its engin

A 150.0 kg rocket moving radially outward from Earth has a...

Key Concepts

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like air drag), the total mechanical energy (the sum of kinetic and potential energies) of a system remains constant. In problems involving gravitational fields, this conservation law allows one to relate the kinetic energy at one point to a combination of kinetic and potential energies at another, making it possible to compute changes in speed or height without directly solving the equations of motion.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion, and it is calculated as one half the product of its mass and the square of its speed. It quantifies the work needed to accelerate an object from rest to its current velocity, and in the context of gravitational problems, it converts into gravitational potential energy as the object moves against the gravitational field.

Gravitational Potential Energy in a Central Field

In a central gravitational field, such as that around Earth, the gravitational potential energy is given by an expression proportional to the inverse of the distance from the center of Earth. This concept is key because it explains how the energy associated with an object changes as it moves closer to or farther from the source of the gravitational field, and it plays a crucial role in determining speeds and maximum altitudes in orbital and projectile motion problems.

Radial Motion in a Gravitational Field

Radial motion refers to motion along a straight line from the center of the gravitational field outward (or inward). In such problems, the analysis often involves the conversion between kinetic and potential energies. Understanding radial motion is essential for determining how an object that is launched directly away from a planet will decelerate under gravity and eventually come to a stop at its maximum height before falling back.

Transcript

00:01 So this question is asking us to determine the kinetic energy of a rocket after going up through the arts atmosphere and beyond, and then determining the maximum height that it reaches given its initial velocity.

00:13 And also the height at which this velocity occurs at.

00:18 So this is basically a conservation of energy question.

00:24 So part i ask is to find the kinetic energy when it's at a height of 1 ,000 kilometers.

00:29 Kilometers so we're given the initial total energy at a height of 200 kilometers so ei is equal to a half m v i squared the initial velocity minus the potential energy of gm m over or which is the height so then this is equal to a half m is 150 kilograms by vi is 3 ,700 meters per second squared and this is minus g, which we know is 6 .7 by 10 to minus 11.

01:06 Mass of the earth is 5 .97 by 10 to the 24 kg.

01:11 Mass of the object is 150 kg.

01:15 Then it's at a height of, it's at a height of 200 kilometers at this height.

01:24 So it's going to be the radius of the earth, which is equal to 6 .37 by 10 to the 6 meters, plus 2 .27.

01:36 Thousand kilometers that's two two by ten to the two by ten to the five two by ten to the five meters as well so then if we calculate this we get the i the initial energy is equal to minus eight thousand eight point zero six five by ten to the nine jules then we also have the final total energy which we need to find the kinetic term of this so this equal to k which we're looking for minus again g m m but this is at a height of at this time it's it's at a height of big oar which is equal to which we're trying to which is equal to a hundred uh raise to the earth plus uh 1 ,000 kilometers so this is k minus so uh this is equal to uh g times the mass the earth 5 .97 by 10 to the 24 kilograms by 1 .150 kilograms mass of the rocket all over raised the earth 6 .37 by 10 to the 6 meters plus the 1 ,000 kilometers that's plus 1 by 10 to the 6 meters so if we calculate this we get that and and so using this if we let ei you for the ef since energy, total energy is conserved.

03:20 Ei is equal to ef, since we're not accounting for friction.

03:23 And we rearrange this, we get that k is equal to, so it's adding this to ei, we get the k is equal to 3 .946 by 10 to the 7 joules.

03:42 So that's our answer for part a.

03:47 Then for part b, again, we use the conservation of energy, so we're asked to determine the, us to determine the maximum height that it reaches...

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