An astronaut is inside a 2.25 × 106 kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4g. (a) What is the maximum initial thrust this rocket's engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket.

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Hey everyone. So today we're dealing with the problem about forces and forces acting upon a body. So we're being told that we have an engineer who has designed a spacecraft with a weight of 6.32 times 10 to the five kg. Now, this vehicle is launched vertically Vertically upwards and the co engineer also decides to accelerate this vehicle as fast as possible all the way up to a speed of 350 m/s. However, we're being asked to remember or we're being asked to be careful that an astronaut in the space vehicle will pass out if the acceleration exceeds four times the force of gravity or four times the force of acceleration due to gravity or forced 4G. Excuse me. So with all of this, we're being asked to figure out what is the maximum thrust generated by the spacecraft's engine at the brink of a blackout. In other words, what is the force due to thrust that the engines will x or that the engines will exert upon the or opposing the weight of the aircraft At 4G. At an acceleration that would push the spacecraft or the space man to the brink of unconsciousness. So, when dealing with forces, the first step that we should always do the first step, it's a draw to force body diagram. Now, let's say that we have our spacecraft right here and we know a few things, the spacecraft is launched vertically upwards. So it is launched in the positive direction, it has a positive acceleration. And if we're doing acceleration to the brink of unconsciousness, right to the brink of a blackout. That means we have a positive acceleration Which is equal to 4G, four times the force of acceleration. Teacher gravity. Now we also have a couple other forces acting upon the uh upon the space vehicle. We have the force due to weight, right? We have the weight which is the mass times the gravity or the force of acceleration due to gravity. And we have the force of trust what we're looking for. We have the force of thrust and that force of trust is propelling the spacecraft upwards. So now that we have our force body diagram, we can go ahead and continue on to our next step, which is utilized Newton's second law and recall that Newton's second law states that the sum of all the forces in a system will be equal to the mass times the acceleration. So if we take up the sum of the two forces right, we have a positive force in the positive y direction and we have an opposing force, its weight which goes in the opposite direction. So if we write that out, we have the force due to thrust minus. They forced you to wait as the sum of all the forces because one is positive, one is in the negative direction, multiplied by the mass, multiplied by the acceleration, which we determined to be alright This in blue, Which we determined to be 4G. With this in mind we can go ahead and moved to the third step, which is actually solving for one of these terms. Right? So we're being asked to solve for the maximum thrust generated by the spacecraft engines at the brink of blackout the force due to thrust at 4Gs of acceleration vertically. So rearranging to isolate the force you to trust. We get W plus four MG. And this comes from expanding. Multiplying mm +24 G. So you get four MG. But if you recall we've stated that weight is also equal to MG, which means this will just be uh MG plus four MG equals C five M G. And we also know what M is the mass of our space vehicle and we know what she is. So let us expand this a little further. We get the sequel to five into Alright this in red 6. times 10 to the 5th kg. That is the mass of the space vehicle, multiplied by the force of accelerator, oops. The force of acceleration due to gravity Due to gravity, which is 9.81 meters per second squared. Which when solving and equating we get the force of thrust generated by the spacecraft's engines at the brink of blackout is 3.10 times to the seventh Newtons or an answer choice B. I hope this helps. And I look forward to seeing you all in the next one

Verified Solution

Key Concepts

Free-Body Diagram

Newton's Second Law of Motion

Acceleration and G-Forces