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$\cdot$ The champion jumper of the insect world. The froghopper, Philaenus spumarius, holds the world record for insect jumps. When leaping at an angle of $58.0^{\circ}$ above the horizontal, some of the tiny critters have reached a maximum height of 58.7 $\mathrm{cm}$ above the level ground. (See Nature, Vol. $424,31$ July $2003, \mathrm{p} .509$ ) (a) What was the takeoff speed for such a leap? (b) What horizontal distance did the froghopper cover for this world-record leap?

4.00 $\mathrm{m} / \mathrm{s}$1.47 $\mathrm{m}$

Physics 101 Mechanics

Chapter 3

Motion in a Plane

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

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So this is a sketch off the project tile off the frog on. We use coordinates with the emerging at the ground and bless y axis upward on DH. We have a X to be called a zero, since there is no acceleration in the horizontal direction. Acceleration long in the vertical direction is negative. 9.8 meters per seconds squared, which is acceleration due to gravity. Andi, We know that at maximum height said this point velocity along why direction and Siegel? Because velocity is always Tandon cheer to the pot. So at this point, and the maximum height velocity is a long exacts is so there is no component of velocity in the sciences and that's ivy. I see. Now we have to find the initial speed on the range off the frog or the distance horizontal distance covered by the frog. So for the first part, But we have to find the initial velocity. So we use kind of medical cushions for that. Andi Well, Eustis in question. So the reason we and using this one is we already know all the variables except, um, we not white on therefore it this equation is easier to solve in the rest off the kind of medical question you have to find time. First on, then only we will be able to solve them. So this one is basically easier to solve because we just have one unknown parameter, which is Levi on DH. Well, just solve for this. Still we not white study. So with this all for me, not quite not quite the way is basically sindoor at the maximum height. It's for that. This is not like acceleration is Oh, that's listen to the gravity on y minus y not just basically watch, but it's just the maximum I Dex going. And this gifts we're not like scared to Quito. Negative off to a vine. Thanks. That implies they not wise square, Rudolph. Negative. Twice off, right and set. And when you solve this, you'll get this answer to be t point 39 double a second Now insolvent This. Make sure that you can vote the height the maximum I had given from centimeter to meet us. Just to be consistent with the units on DH. I notice that everyone is negative. So negative times negative is positive. So, uh, you know not why you won't be a complex value now next, Like this in dance off, we not on the angle. So we're not Why is basically be not times sign off the angle it makes with the horizontal. And so this we found of the three point 39 beautiful second on. Since we have the angle given just 50 degree, we can find me not It will be called to keep 1390 ver sign off this angle and we get this. Val, you Toby. Four point noodles. Me too for a second. So this is the value ofthe the initial speed off the Franck now, for the second part is to find the change. So really was kind of a cushion for the Y axis of study for the X axis. So it's big a question. There you have the James involved, the acceleration on DH the time. So for that first we need to find the Times. For that, we will lose our question involving the like on opponents. So we're basically solved the kind of medical question in the vertical direction. And since they're not, why citizens since I was the maximum height Ah, the destined okay? Yeah. So we're solving for the time to total time taken by the frog toe. Go from here to here. So are my Linus. Why not? Will basically be zero? Because it's returning to the same point in light addiction so free from here again. Right? So take that in common and we have off a light. Good to see you on. This gives one solution, which is Steve's going to see your on another solution is is equal to twice of negative off myself did not like All right, a right. So take was to zero basically corresponds to the point. From there, the fog starts and includes two. This values the time that the frog reaches the other. And so this one. So this is basically the time ofthe dragon. So we will solve this and we get this time to be quiet. Cedar point 692 seconds. So how does that evil is? Negative. So your negative and miss negative sign cancels out on either the positive value for time now, we will use this time, uh, in the kind of medical question for the horizontal direction to get the value of this range. So we will use the same question as this one that will solve this for the horizontal direction And since acceleration along extra addiction on along the horizontal direction cereal. So this term goes away on this value Over here is the rains that we want to find. So this gives our Toby called toe Reno text time, Steve, and being addicts is basically living not times call Sign off the Andal We not mix with the what is undone And today is the time off light Still, the planters are losing to the question and we get this change to be 1.47 Don't now I never easy me Not easy, but basically on the ultimate solution to finding the time could have been solving for they not so we know that three light is going to be not Dwight, plus the righty on at the highest point. I mean, why is cereal right? So they get Toby Quinto negative off. They're not by over a lie, but this time is just, uh this day is just the time taken by the frog toe reach the maximum height. So to cover the whole part, the time should be prised off this value. That means the total time. So let's call the state dash And this tea when big guys off this it is dry. So we not right over here. It is the same result as the god of a shell. So it's up to you too. They quotable We question you wanna use to find the time.

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