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That's a lot of hot air! A hot-air balloonist, rising vertically with a constant speed of 5.00 $\mathrm{m} / \mathrm{s}$ releases a sandbag at the instant the balloon is 40.0 $\mathrm{m}$ above the ground. (See Figure $2.52 . )$ After it is released, the sandbag encounters no appreciable air drag. (a) Compute the position and velocity of the sandbag at 0.250 s and 1.00 s after its release. (b) How many seconds after its release will the bag strike the ground? (c) How fast is it moving as it strikes the ground? (d) What is the greatest height above the ground that the sandbag reaches? (e) Sketch graphs of this bag's acceleration, velocity, and vertical position as functions of time.

a) $=5-2.45=2.55 \mathrm{m} / \mathrm{s}$$v(1.0)=5-9.8=-4.8 \mathrm{m} / \mathrm{s}$b) $t_{s t}=3.41 \mathrm{s}$c) 28.4 $\mathrm{m} / \mathrm{s}$d) 41.27 $\mathrm{m}$

Physics 101 Mechanics

Chapter 2

Motion along a Straight Line

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

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So let's take. Was it the right of the upward on DH? Was directs to the right on DH. Therefore, if we have acceleration in the vertical direction to the negative 9.8 make up a second scotch. The initial velocity of the sandbag equals the velocity off the balloon. So we not wine five meters per second. Also in the balloon Just a ground. Its height is Ryan Linus. Why not? Riches 40 point negative 40 point Ciro. So basically, when you treat just the ground, why become zero the height from the ground? Because ground is basically the base on the origin on why not? Which is the height off The balloon from the ground initially is 40 meter. So my minus, why not gives negative 40 meter Also at the maximum height, the sandbag has the velocity v Y record to zero. Let's do But if now so let's solve for position and velocity at sit off point do five seconds So they will use this a question to solve for my Linus ly not for the sand back. So this is equal given the values they should be equal to. I see no 0.94 meter. Make sure while solving this will include this negative sign off acceleration. So why minus Why not is now 3.94 meter? Therefore, the sandbag. The position off the sand back from the ground is 0.94 meter, plus the height off the balloon from the ground initially, which is why not, which is equal to 40 meter and therefore assigned by is 40.94 meter from the ground. Now let's find the velocity for dad revenues This equation over here on DH, this's equal do to point 55 meter music. Now we will use exactly the same question to find the position and velocity at one seconds. So basically, we will use this a question over here and also this a question. But for equal the one seconds so we know that they get my minus. Why not Toby Syria, 0.1 meter on velocity could be negative 4.8 meters per second. And since we get my minus y notto 0.1 meter Ah, the sand back is basically this will do the same that we did here said The sandbag is 40.1 meter from the ground at this time for the second part. So we have eye minus find out. Toby. Negative. 40 meters. We also have we not rightto be five meter per second and just rewriting the questions that I wrote here that's given we have a light to be negative 9.8 meters per second squared. Now we used this equation basically this one over here to solve for time. And as you can see, this will be quite a ticket question. So we solve this time for thiss distance to travel this amount of distance on DH, we get time to be equal to sit a 0.45 plus minus two point 90 seconds. So this will give one positive value one positive solution and one negative solution. But since that time should be positive with picked a positive solution which is 3.41 seconds fiscal 0.51 plus 2.9 incidents. Now, if you wanna know how to solve the quadratic equations, let me just a light agenda formula for that. So this is how it quite a dicky question looks like on DH. We saw it for next and for that this is the question that you should solve. For it has two solutions. One for less and one for taking a minus. Sign on over here. If you take my minus. Why not to the right and side Or, um, this dumb or to the left Can sites are basically making one side to be equal to zero. And the other side looked like, Ah, this term over here. You see that? Yeah. Ex corresponds to team. Your is basically on the coefficient ofthe express, which is half off a white you'll be is coefficient ofthe X. It is we not life and C is the constant which will be negative off my minus lineup because it goes to the other side. So the sign will change now. Therefore, this is why not minus y on. So we use thes values, plugged them into this equation. Tau. Find time on DH. We'll get two solutions. One for less, one for minus. And we picked the positive luncheon since time should be positive and it comes out to be 3.41 seconds. No. When the tide bite, we have to find the final velocity. So given the value's off. Not why a right on DH this time. We will solve this equation and they find this velocity to be negative. 28. Find Ford meet over a second. This means that the lost velocity has a magnitude 28.4 meters per second on points downwards with the last part. So for this case, we have we not y to be equal toe. Five meter per second Acceleration is again negative. 9.8 meters per second staff Final velocity should be Cedo now the salt for ah, the maximum height where the rate is zero. So this is the question that we should solve for which involves why minus why not on the perimeters. Oh, we are already given the value's off. The parameters were already given. So this is a question that we should use. This gives my minus Why not Toby Equal toe negative B not lice that over to fight. Given these values, we solve this and we get this value. Toby, one point a to eight meter on DH. This so this is the height off the sandbag. But since the balloon is 40 meters from the ground, therefore the maximum height will be 1.2 weight plus 40 and therefore for divine point, you ate meat from the bone. This's the maximum might with the last part. We have to draw graphs off anyway. Force's sti half off. They live our Sistine on DH. Why versus steep now since acceleration in our kisses constant on its negative basically points in the negative i direction So to the point, you should be in the negative on the fourth quadrant and should be constant. Let me just pick another column. This is the blood. This is the origin now for velocity Final velocity. If you look at this equation over here, you see that velocity is a linear function ofthe time. So basically, you know that Neary question looks like why equal to m explicitly where m is the slope and C is the y intercept. So if you compare this equation tau this accretion, we'll see that you're slow is a Y on the X intercept is Venus the Why intercept is we're not like so we do the same over here. So this will give ups now, since acceleration, which is our slope, is negative, Andi initial velocity has a positive value. Therefore, it should start from a positive value on DH. If acceleration is positive, which is the slope we should go like this. But here acceleration is negative. So it should have a negative story. And therefore we should go like this and this value over here this be not right and slow off. This graph is a light musically the used This is Christian do alive. Went over here now. But why? We destroyed the question, JJ. Why connecting relating my auntie. So is this a question is the one that we will lose on? As you can see, this is quadratic, right? Because we compared this the desecration to solve for X or time. And so this is a quadratic equations of the graph the block off. Why voices, Steve, then be if Why not? It's non zero. So that should start from a point. Why not? We should go like a parabola. Right? So this is the graph for y versus T taking the final. Why is single at the ground

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