Parachutists whose chutes have failed to open have been known ...

Question

Parachutists whose chutes have failed to open have been known to survive if they land in deep snow. Assume that a 75-kg parachutist hits the snow with an area of impact of .030m² at a velocity of 55 m/s, and that the ultimate strength of body tissue is 5 x 10⁵ N/m². Assume that the person is brought to rest in 1.0 m of snow. Show that the person may escape serious injury.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem, dealing with stress and ultimate strength in this problem. A person with a mass of 85 kg accidentally falls off of a helicopter into a haystack. The area of impact was later found to be 0.25 m squared and the velocity at which the person landed was 45 m per second. If he was stopped after traveling a distance of 1.5 m into the haystack and the ultimate strength of body tissue is 4.6 multiplied by 10 to the five newtons per meter squared. Evaluate if he, if he can escape significant injuries, we're given four possible choices as our answers. Voice A he can escape significant injuries but minor injuries are possible. Voice B, he will certainly suffer significant tissue damage. OC He will certainly be dead od no conclusion could be drawn due to insufficient information. So since we need to um compare whatever we find with the ultimate strength of the body tissue, what we need to actually find is the stress that's going to be exerted on the person by the haystack. So we call our definition of stress. So stress is equal to the force divided by area. So stress is equal to F divided by A. So that is gonna be the force that's being applied over the area. A now we were given the area that was the 0.25 m squared, but we don't have any information about the force. The only thing that we were given is the speed at which the person landed and the distance that they traveled. And since we have a speed in the distance here, we want to actually use energy conservation. And what we'll actually um use is the work energy relationship. So that if you recall that it is the total work being done is equal to the change in kinetic energy. So here, the only work that's being done is that um of the haystack acting on the person and recall our definition of work that is just gonna be the force multiplied by the displacement and will then multiply it by the cosine of the angle between those two vectors. Now, in this case, we have the force going upwards and the displacement going downwards. So those two vectors are going in opposite directions. So the angle between those two forces or those two vectors is 100 80 degrees. So that quantity has to be equal to our change in connect energy. So that'll just be my final connect energy minus my initial connect energy. So we'll have FD cosine 100 80 degrees is equal to KF minus K I, now the cosine of 100 80 degrees is negative one. So I will have on the left hand side minus FD and that's gonna be equal to, for my final kinetic energy, I'm gonna have zero. That's because my person is gonna be stationary at the end. And so therefore, there's going to be no connect energy. I will still have my initial connect energy. So I'll have minus and recall our formula for connect energy that is one half MV squared. So it'll be one half M the I squared. And here the V I, here, the initial velocity or initial speed is the speed at which the person lands on the haystack because that's when the person is going to begin to slow down. So we're interested in the force. So let's solve this for the force. So we'll have F is equal to negative science cancel. And I will have MV I squared provided by UD. So I can now take that force and plug it back into my stress equation. That means that my stress gonna be equal to, I'll have MB I squared divided by two D A. Now I have values for the mass, the speed, the displacement and the area. So I can just plug in values at this point. So I have the stress equal to for the mass, I have 85 kg or the speed. That was the 45 m per second. So I'll multiply the 85 kg by 45 meters per second. And that quantity gets squared, they'll divide that by two. The displacement was the 1.5 m. I have 1.5 m and that gets multiplied by the area which is the 0.25 meters squared. And so when I calculate my stress, this gives me a value of 2.3 multiplied by 10 to the five Newton's per meter squared. So what I need to do now is compare this to the ultimate strength of the body tissue. And we were given that value that was the 4.6 multiplied by 10 to the five newtons per meter squared. And so by 2.3 multiplied by 10 to the 5 m per meter squared is less than the ultimate strength of the 4.6 multiplied by 10 to the five mutants per meter squared. That means the person is going to escape any significant injuries since it's about half that value. However, we're assuming that the force is applied equally across the 0.25 m squared area of impact. And there could be certain areas um since the body is not a flat surface, there could be areas that have a little bit higher stress value, you could see some minor injuries there. So if we look at our answer choices, this corresponds to answer a. So just a quick little recap. Um Since we needed to compare um our values to the ultimate strength of the body tissue, we need to first calculate the stress that was um exerted during this impact. And in order to calculate the force, we needed to actually use the work energy theorem and our definition of work and connect energy along with that once we are able to find the force using that expression, we could plug it back in and calculate a value for our stress. So I hope that this has been helpful and I'll see you in the next video.

Key Concepts

Force and Impact

Pressure and Area

Energy Dissipation

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