When jumping, a flea accelerates at an astounding 1000 m/s^2, but over only the very short distance of 0.50 mm. If a flea jumps straight up, and if air resistance is neglected (a rather poor approximation in this situation), how high does the flea go?

Video transcript

Hey everyone in this problem, we have an insect escaping a predator attack that produces a straight vertical acceleration of 400 m per second squared over an extremely brief distance of two centimeters, were asked to find the maximum height reached by the insect during its escape maneuver and we're told that we can ignore air resistance. Now when we think about this problem, we're gonna need to do this in to kind of stages. Okay, So the first stage is when that insect produces their own vertical acceleration of 400 m per second squared. Okay, so they're going to produce that acceleration, they're gonna rise only for two centimeters. But at the end of that acceleration, they're still going to have an upward speed. They're gonna travel higher than that two cm before the acceleration due to gravity brings them back down. And so we're going to have the first stage where they have that vertical acceleration of m per second squared. And then we're going to have a second stage where the acceleration is going to be just the acceleration due to gravity acting downwards. So we're gonna take up to be our positive direction. And in stage one What do we have? Well, the initial speed is going to be 0m per second because the insect is gonna start from rest. The final speed. In that first stage, the speed that the insect reaches at the end of that vertical acceleration period, we don't know The acceleration we know is 400 m/s squared. And the distance delta. Why that they travel is 2cm and we're gonna convert this into meters. So we're gonna multiply by one m per 100 centimeters. Okay? So to go from centimeters to meters, we divide by 100 oh yeah 0.2 m. And we aren't told any information about the time either. Now we have three things that we know. So we can use our you am equations and we have to figure out do we want to find out VF or do we want to find the time T? And what we want to do is we want to find the time V. F. Because that's gonna tell us our initial speed for the second stage. Okay. If we know the speed that the insect reaches when the acceleration of 400m/s is over, we can figure out what's going on after that point. So we're gonna choose the um equation that doesn't include time and that's going to be the following the F squared. And we have subscript of one on all of these values just to indicate that we're talking about the first stage. So we have V. F one squared is equal to v not one squared plus two. A one delta Y one. Now V. F. That's what we're trying to find. That's equal to V not square, that's gonna be zero squared. So we just get zero there two times the acceleration, 400 m per second squared times delta. Y one, which is 0.2 m. This is gonna give us a V. F squared is equal to 16 m squared per second squared And so v. F one is equal to four m/s. So we know the speed now that the insect is going when their acceleration stops. Okay, so now we can look at stage two and that final speed from stage one is actually going to be the initial stage or initial speed, sorry, for stage two. So in stage two the initial speed v not to Is going to be the final speed from stage one. Okay. Which we found to be four m/s. We don't know what the final speed will be. We know that the acceleration is going to be negative 9.8 meters per second squared and we want to find delta Y two. Hey we want to know the maximum pipe. Now if we're thinking about finding the maximum height, we want delta Y two. When we're at the maximum then we actually do have information about the speed. V. F. Okay, the speed V. F. Is then going to be zero because at the maximum height the insect is going to stop momentarily before it starts to fall down. Okay, when it changes direction from upwards to downwards momentarily it's gonna come to rest, that speed will be zero And so Delta Y two will indicate how high they get at that max. Okay, so again we want to use the equation that doesn't include T. We don't have information about T. And we don't want to find information about T. Or it's not needed for this problem. So we're gonna choose that same equation, the same um equation as from the first stage where V F squared is equal to V not squared plus two A delta Y. So we have that zero is equal to four m per second, all squared plus two times negative 9.8 m per second squared times delta Y two. and so Delta Y two is going to be equal to, we get four m per second squared is gonna give us 16 m squared per second squared. We're gonna move this to the left hand side. And so it's gonna become negative 16 m squared per second squared. And in the denominator we're going to divide by two times negative 9.8 m per second squared. So we get negative 19.6 m squared per second squared. And this gives us a Delta Y two of 0.816 m. All right. Now we have to be careful. We can't just answer with eight point or 8.0.816 meters. Okay, that can be really enticing to go and answer the problem like that. However, remember we're trying to find the maximum hype And this was a two stage problem. So the distance we traveled in stage two was . m but we also traveled two cm in stage one. And so in order to find our maximum height, we actually need to add the two together. So the maximum hype Is going to be equal to Delta Y. one Plus Delta Y two, which is equal to two centimeters plus 0. m. And now if we look at our answer choices, we see that they're in centimeters. Okay, so let's convert everything to centimeters instead of two m. So this is 8.816 m. We're gonna multiply by 100 centimeters per meter to turn this into centimeters and that's gonna be two centimeters plus 81.6 centimeters for a total Of 83.6 cm. And so our maximum height is therefore 83. cm for that insect. And if we go back we look at our answer choices. We round to the nearest centimeter, we see that we have answer choice. C the maximum height reached by the insect is 84cm. Thanks everyone for watching. I hope this video helped see you in the next one.