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. (b) What force, in terms of the astronaut's weight w, does the rocket exert on her? Start with a free-body diagram of the astronaut.

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Accepted Answer

Hey everyone. So today we're dealing with the problem regarding uh, forces and forces acting upon a body. So we're given a lot of things. So let's sort of break this down. Look at this question stem. So we're told that we have a space vehicle design that weighs or has a mass of 1.28 times 10 to the fourth kg. And this vehicle is launched vertically upwards And a co engineer decides to accelerate the space vehicle as fast as possible to a speed of not 290 m/s. But if this vehicle exceeds for Gs, the pilot India in the space vehicle will pass out. So we need to we need to determine what is the force exerted on the astronaut by the space vehicle in terms of the astronauts, wait w so let's go ahead and draw this out. Let's say we have our space vehicle and I'll drive over here, we have our space vehicle. Let's try it like so, right, no, we're told a few things, we're told that it has an initial upwards velocity and the positive y direction. This is its initial velocity. It's accelerated vertically upwards. Right. The Co engineer also accelerates accelerates. Oops, Co engineer also accelerates it vertically upwards as mentioned right here, it's accelerated as fast as possible. No, we also know that in the space vehicle, there's the weight of the astronaut plus the vehicle itself downwards as well as the normal opposing force. It'll be upwards. So, when working with forces like this, we draw a force body diagram as it really helps us sort of conceptualize what's going on. Our next step that we need to do because we need to recall Newton's second law, right? And Newton's second law states the sum of all the forces will be equal to the mass into the acceleration. Right? So let us consider what are the forces acting here? Well, we have the normal force going upwards and it's opposing force the weight. Right now, we know that there is a mass and we need to make sure that the acceleration does not exceed for Gs, right? Because again it cannot exceed for Gs with this, we can now solve for end because we need to find the force again. We need to find the force exerted on the astronaut by the space vehicle and that would be that would be the normal force opposing the weight, pulling the vehicle down. So rearranging Step three, solving, let's just write this is step two and this is step one. Just to clarify, Step one is rather fourth body diagram. Step two is write it out in Newton's second law form and then step three is to solve So rearranging if we're solving for the force exerted by the space vehicle, which is N. We get em into four G plus W. Which can also be written as W plus four MG. Now, if we recall weight is nothing but MG right? It is the mass times the force of acceleration due to gravity. So this means that this will be we can substitute this in and we have M. G plus four MG, Which is equal to five MG. Let's go to five MG. Similarly, five MG is nothing but five W. Because again, five MG is five W. So The astronauts or the force exerted on the astronaut by the space vehicle will be answer choice D five W. 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 due to Gravity