On September 8, 2004, the Genesis spacecraft crashed in the Utah desert because its parachute did not open. The 210-kg capsule hit the ground at 311 km/h and penetrated the soil to a depth of 81.0 cm. (b) What force did the ground exert on the capsule during the crash? Express the force in newtons and as a multiple of the capsule's weight.

Video transcript

Hey everyone today, we're dealing with the problem about Newton's second law. So we're being told that a 140 kg drone, 140 kg starts falling because it ran out of fuel at high altitude Due to having no safety measures, it falls to the earth and crushes and hits the snow at 278 km/h and goes as deep as 68cm with this, we're being asked to find the force exerted on the drone by the snow and that's very important by the snow during the crash in Newtons and what that force is as a multiple of weight. So let's break this down. So, to work through this problem, there are three things we need to do. We need to first draw a force diagram with the force diagram. We can then go ahead and substituting values into the Newton's second law, which states that the sum of all the forces within a system is equal to its mass times acceleration. So that is the net force is equal to mass times acceleration. And finally from there we can solve. However, there's a few things we need to do before even getting to the Newton's second law before getting to mass times acceleration in the first place. And the reason for that is even though we're given the mass, it's 140 kg we don't know the acceleration and to solve for acceleration. We actually need to utilize some Kinnah Matic equations. So, for vertical kinda Matics, because something is falling, we're dealing with the Y direction. If we have to think about it as a graph, we need to know a few quantities, Let me write these out in blue, we need to know the change in vertical height as in and this is the change in vertical height from when it hits the snow and how far it penetrates into the snow. We need to know the final velocity, the final velocity in the Y direction. We need to know the initial velocity in the Y direction. We're trying to solve for acceleration in the Y direction and it's just acceleration period because we're not concerned with horizontal aspects at this moment and we're not given time. So let's think about this. Well, it says that it penetrates the drone penetrates the snow at a depth of 68cm. We want this in meters because generally we want everything to be in there uh Based SI unit, sorry. So 68 cm, we can recall that one m is equal to 100 cm sort of centimeters will cancel out. And An important thing to notice it penetrates at a depth of cm, which means it goes, hits the snow and goes further down. So it goes a negative 68 cm. So converting this, we get negative 0.680 m. The final velocity will be zero m per second. And this is because we are concerned with the force exerted on the drone as it falls or as it hits the snow and penetrate into it. So when it finally comes to a stop, that's its final velocity, it'll be three m per second. The initial velocity Is given to us as 278 km/h. But we need to find this in m per second. So let's do some conversions and I'll do them down here. We take km/h. We know that we have 1000 m for every one km. We know that we have one within one hour we have 3600 seconds. So canceling out our respective values ours will also cancel her. We will get 77.2 m/s. So the initial velocity is 77.2 meters per second. However, where the velocity, the initial velocity will be negative because the object is falling, it is going in the negative y direction. This is our target, the acceleration, this is our target. So this is what we're trying to solve for. And now that we have three out of the five values 12 and three, we can go ahead and use one of the universally or sorry, not universally uniformly accelerated motion equations. That doesn't include the unknown T. And that equation is as follows, it is that the final velocity is equal to the initial velocity plus to a times the change in distance or the change in vertical distance. So two, sulfur A. We can seven. All the other values, zero is equal to negative 77.2 m per second squared Plus two. We're trying to solve for a multiplied by hoops, multiplied by -0.680 m. And this means that A. Is therefore equal to 4.38 times 10 to the sorry, not negative 3rd 10 to the third meters per second squared. So this is the first part. We now have a we now have our acceleration and it is a positive value which means it is going in the positive Y direction. I'm gonna scroll down a bit so we have a little more space. So just bear with me for a second. So now that we have this, let's go ahead and actually draw a free body diagram to illustrate the actual path of motion. So if we have our drone which I'll designate as so the force pulling it down the force downwards as the mass times the force of gravity which is the same thing as the objects weight as it hits the snow. However, there will be a normal force exerted a normal force force normal and we as we can recall as we just found out there is also a positive acceleration. So because there is a positive acceleration but a downward velocity, the object is slowing down and that makes sense because it's hitting the snow and then it comes to a halt. So with this in mind let's go ahead and actually start writing out our Uh Newton's 2nd law, our force is equal to mass times acceleration. So it's the sum of all forces. The Y direction. Because we're only dealing with vertical motion is equal to the mass times the acceleration in the Y direction. So rewriting this the sum of the forces here, there's a normal force acting up and there is a weight force that is in the Y direction or in the negative Y direction. So the normal force rewrite that the normal force, the normal force minus the weight will be the total net force is equal to mass times acceleration. From here, we need to find the normal force because again, the question is asking us to solve for the force exerted by the snow, which is indeed the normal force. So isolating the normal force gives us weight plus mass times acceleration. And remember weight is nothing but mass times the acceleration due to gravity or G plus mask. And I leave that. Why now? Which means we can factor at the M and we get em multiplied by a quantity of G plus A. So from here we can go ahead and start solving substituting in our values. So you have that the normal force is therefore equal to The mass is 140 kg, which I'll keep in black 140 kilograms multiplied by The force due to gravity is 9.8 m/s squared. And the acceleration as we deduced earlier. Excuse me, is 4.38 times 10 to the third meters per second squared. So we get an answer Of 6.1515 times 10 to the 5th Newtons. So that's the force exerted by the snow. But we're still not done. We are asked to find what this value is as a multiple of the weight. Now, this means we need to take the normal force. The normal forest divided by the weight, which is equal to 6.15 times 10 to the 5th. Newtons over mass times gravity. Mass times the force of gravity rather because weight is MG. Which means we can simplify this further or expand it. Sorry, we expanded further. 6.15 times 10 to the fifth. Newtons divided by 1 40 kg, Multiplied by 9.8 m/s squared. Which gives us a final answer of 448. So this means that the normal force Is therefore equal to 448 times the weight. So looking back at her answer choices if the force exerted by the snow is 6.15 times 10 to the 5th Newtons. And The force as a multiple of the weight is 448 times the weight that gives us answer choice a I hope this helps. And I look forward to seeing you all in the next one.