An 80 kg astronaut has gone outside his space capsule to do some ...

Question

An 80 kg astronaut has gone outside his space capsule to do some repair work. Unfortunately, he forgot to lock his safety tether in place, and he has drifted 5.0 m away from the capsule. Fortunately, he has a 1000 W portable laser with fresh batteries that will operate it for 1.0 h. His only chance is to accelerate himself toward the space capsule by firing the laser in the opposite direction. He has a 10 h supply of oxygen. How long will it take him to reach safety?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A 65 kg geologist on Mars is stranded 18 m away from her rover after a rock slide due to the steepness she is unable to climb, she uses her 1800 watt laser which has 3.0 hours of battery life to generate thrust by firing it downwards, lifting herself out. She has a 18 hour supply of oxygen. How long will it take for her to reach the rover? So that's her end goals. We're trying to figure out considering her situation, how long it will take for her to reach the rover, given these specific conditions? Awesome. So we're given some multiple choice answers. They're all in units of hours. So let's read them off to see what our final answer might be. A is 3.5 B is 5.0 C is 6.2 and D is 2.1. So first off, let us write down all of our known and unknown variables. So we know the mass of the geologist is 65 kilograms. We also know the distance of the rover is 18 m. We also know the power of the laser which is equal to 1800 watts. We also know what the speed of light is. We know that the speed of light is 3.0 multiplied by 10 to the power of 8 m per second. But we do not know what s will be. We do not know what or S3, we do not know the distance traveled in three hours. We do not know what the acceleration is. We do not know what the final velocity will be, but we do know what the initial velocity in this case and she's not moving, there's no motion, it'll be 0 m per second. So we know the initial velocity. And finally, the last value that we do not know is we do not know the time it takes for her to reach the rover. And this is our final answer. OK. So in order to find our final answer, we need to find a couple other of our missing variables first, in order to help us solve for how long it will take for to reach the rover. So to do that, let us now recall and use the equation to determine the power emitted by the laser by relating it to the force equation. So that states that P the power is equal to the force multiplied by the speed of light. So let's rearrange this equation to solve for F. And when we do that, we'll get that F is equal to power divided by the speed of light. So now we could plug in our known variables and SOLF for F and when we do that, we'll get that F is equal to 1800 watts divided by the speed of light, which is 3.0 multiplied by 10 to the power of 8 m per second. So when we plug that into a calculator, we will get 6.0 multiplied by 10 to the power of negative six newtons. So now at this point, we need to recall and use Newton's second law to solve for the acceleration. And let's recall that, that states that F equals M multiplied by A. So force is equal to mass multiplied by acceleration. So, and we need to rearrange this to solve for A and when we do that, we'll get that A is equal to F divided by M. And now we need to plug in our known variables. And so for A and when we do that, we'll get that A, our acceleration is equal to 6.0 multiplied by 10 to the power of negative six newtons divided by 65 kg, which will give us that the acceleration is equal to 9.23 multiplied by 10 to the power of negative 8 m per second squared when we plug it into a calculator. So now we need to recall and use the first equation of motion which states that the final velocity is equal to the initial velocity plus the acceleration multiplied by the time. So note yet again, you mentioned it before, but note that the initial velocity is equal to 0 m per second. And in this case, t we're trying to figure out the lifetime of the battery. So we need to simplify and rename T as the lifetime of the battery. So when we do that, we will get that the final velocity is equal to the acceleration multiplied by the, the, the battery life or the lifetime of the battery, which is T subscript L. So now we must quickly convert hours to seconds. So let's quickly note here that three hours is equal to 10,800 seconds. So you can quickly get this conversion by using dimensional analysis or you can just look it up. I've just gone ahead and done the conversion for you. So this is equal to our T lifetime value. So now we can plug in our known variables and solve for the final velocity. So the final velocity is equal to 9.2 three multiplied by 10 to the power of negative 8 m per second squared multiplied by our lifetime of the battery in seconds, which is 10,800 seconds. So when we plug that into a calculator, we will get 9.9 seven, which we rounded to two decimal places. Because when you type into a calculator, you'll get like 9.968. So we'll just round it to 9.97 multiplied by 10 to the power of negative four meters per second. So now we can, so the distance traveled within the first three hours. So to do that, we will get use the equation that says two multiplied by the acceleration multiplied by the distance S is equal to the final velocity square minus the initial velocity squared. So now we can rearrange this equation to solve for S and and let's keep things simple. So we don't get confused, let's call the distance S subscript three. Since we're trying to find the distance traveled within three hours, and this will be equal to when we rearrange the soul for the distance by itself, the final velocity squared minus the initial velocity squared all divided by two multiplied by A. OK. So we can now plug in our known variables and solve for our distance traveled in three hours which we can write as S subscript three is equal to 9.97 multiplied by 10 to the power of negative 4 m per second squared or this is a squared. So 9.97 multiplied by 10 to the power of negative 4 m per second parenthesis squared minus our initial velocity which was 0 m per second. And this value is squared all divided by two, multiplied by our acceleration, which was 9.23 multiplied by 10 to the power of negative eight meters per second square. So when we plug that into a calculator, we will get 5.38 m. Thus, the distance that the geologist must cover will have to be 18 m minus 5.38 m, which is equal to 12.6 m. So now we could start to solve for our final answer by recalling and using the second equation of motion equation which states that s is equal to the initial velocity multiplied by the time plus one half multiplied by the acceleration multiplied by the time squared. And note that as a equals 0 m per second squared, we can write that S is equal to the initial velocity multiplied by the time which we can rearrange and solve for the time. So the change in time is equal to the distance divided by the initial velocity. So now we could plug in our known variables in salt. So delta T, the change in time is equal to 12.6 m divided by 9.97 multiplied by 10 to the power of negative four meters per second to the power of in negative one, which we can write as delta T is equal to 12637.91 seconds. Which is approximately equal to 3.5 hours. OK. So just for fun, how about we quickly do the unit conversion using dimension analysis, dimensional analysis to solve for our final answer. So it's convert seconds to hours. So 12637.91 seconds. So note that in one minute, there is 60 seconds and that in one hour there is 60 minutes. So as you can see the seconds cancel out the minutes, cancel out, leaving us with hours. So that's how we get 3.5 hours as our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be the letter A 3.5 hours. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Newton's Third Law of Motion

Conservation of Momentum

Kinetic Energy and Work

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