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$\bullet$$\bullet$ Mars Exploration Rover landings. In January 2004 theMars Exploration Rover spacecraft landed on the surface ofthe Red Planet, where the acceleration due to gravity is 0.379what it is on earth. The descent of this 827 kg vehicle occurredin several stages, three of which are outlined here. In Stage I,friction with the Martian atmosphere reduced the speed from$19,300 \mathrm{km} / \mathrm{h}$ to 1600 $\mathrm{km} / \mathrm{h}$ in a 4.0 min interval. In Stage II, aparachute reduced the speed from 1600 $\mathrm{km} / \mathrm{h}$ to 321 $\mathrm{km} / \mathrm{h}$ in$94 \mathrm{s},$ and in Stage III, which lasted 2.5 $\mathrm{s}$ , retrorockets fired toreduce the speed from 321 $\mathrm{km} / \mathrm{h}$ to zero. As part of your solu-tion to this problem, make a free-body diagram of the rocketduring each stage. Assuming constant acceleration, find theforce exerted on the spacecraft (a) by the atmosphere duringStage I, (b) by the parachute during Stage II, and (c) by theretrorockets during Stage III.

Physics 101 Mechanics

Chapter 5

Applications of Newton's Law

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Cornell University

Rutgers, The State University of New Jersey

Hope College

University of Sheffield

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

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question 75. Um, it's a very long question. So I'm not gonna quite read it all but I outlined here. We have a rocket, um, rotering us that's landing on Mars, which I'm representing about this rectangle here. So as it enters Mars atmosphere, it enters very, very fast with this initial speed of 1900 19,300 kilometers and in various stages deemed stage one, stage two in stage three slowly reduces its initial high velocity finally or two on things. 30 of Los E, where it will eventually land on the plan is our goal for each part. Well, hopefully, question is to find the force exerted on the spacecraft in each of these stages. Um, and while their methods to slow down from their initial to final velocities, um, are different, the idea for how you treated her solve would be the same. So we're told initially the mass of the rocket and the acceleration due to gravity on Mars. And we can use that to find the net force applied to the rocket, keep same rocket. But it is it's a rover, So yeah, So if the question that has to draw a free body diagram for each change. Really, it's the same thing happening in each scenario. Um, where the the idea go is to slow down the rocket. The rockets are overt, just so lands perfectly on Mars, so I'll draw one. And really, it's the same for each scenario. So another rock Rover has a weight pulling itself down towards the Martian surface, and it is traveling downwards. So the Net force is the M A down that way, and that is a counter act by the thrust. Emotional was called F here, pulling the like. It's really thrusting it up to reduce the direction of of its velocity so called downward motion to be positive. My positive line. Just be a little easier with death. So I need to find the overall force that would be equivalent to M A plus the weight of the rock Rover man. Keep saying over Rocket. I'm not sure why I suggest you. It's basically forgive me, but I see rocket or older. I, of course, mean the same thing two recent Here we don't have acceleration. So during a stage full sulfur acceleration and use it in this equation where we're going to simplify this to be mass times acceleration plus Argie prime our new G value. Since the weight is just mass times the g of the planet they're doing up And so we can find Tom, However lost our acceleration just by simple mathematics we have for traveling downwards are positive. Are velocities positive down, positive down but work suffering upwards to slow down the motion. So that's why we have a negative sign for acceleration there. If you look, we are given lost initial velocity, final velocity and time so we can simply rearrange this equation to be initial velocity of my spinal velocity over time and for the scenario, I were told velocity in terms of kilometers per hour, which I simply converted two meters per second because that's obviously more appropriate measurement of Yarmuth. Um, these calculations so using are given values for velocity and time. I can plug him into this equation. Here we find that acceleration. What would be negative with me? Oh, no, I would not leave it. Sorry. He's already slowing down. I took the direction into account. The not isn't is 5000 and these 400 on the order of its of course would be positive. But my polity soakers I thought I'd be negative because of the signed edge of my directions. But I intentionally chose it so that we are accelerating. Um, downwards. Right. So excavation is with this 20.5 meters per second and we can find them that forced to be again with the equation stated above 827 kilograms 20 times the acceleration. Plus what we have on, um RG value for this planet 200.379 times are normal value for G. And by playing this in, you find the force in this near 23 significant figures to be 2.0 times 10 to the four news. So without performing the calculations, I significantly expect that force Applied age Nereo, um isn't to drastically high. It's just so if I mean the rover. But ideally, you want to have to break anything in there. So you do want everything to be as steel is possible. But, um, nothing too too intense for the system to handle. So again, I need to find the acceleration of this which this stage I should say, which again is the same part A. Our stage wanted to stay in the same for stage three. So the over tea pavilion, what is plugging in each of the different values for each scenario? So by doing this for stage two with Part B. The question we find the acceleration to be 3.77 meters per second squared and while radically new, the stage three or this time our final velocity is just zero in our celebration is very intense, based on which makes sense, because in this part, why much of what I said it before it. This does reveal, um, retro rockets they call to slow down the rocket. Even more are there. There are over even more So I do expect this because they issue it's a parachute. His day street are due to rockets. The deceleration. So it does make sense that that the acceleration the stage is much, much higher than that could be Chief front is the parachute. And finally we can find the force from each these news. Well, we're doing difference. Here is the term a this one here, everything else remains the same. Someone calculating it again. Three significant figures. The force, due to Stage two and the parachute is he unless than the rockets, which is very, very reasonable in the for centuries. 3.26 times 10 to the four. It's thinking so, recognizing that these the map, the force and may and wait, contributing the same directions and give the forces as follows.

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