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Problem 49 Hard Difficulty

A hot-air balloon is rising upward with a constant speed of 2.50 $\mathrm{m} / \mathrm{s}$ . When the balloon is 3.00 $\mathrm{m}$ above the ground, the balloonist accidentally drops a compass over the side of the balloon. How much time elapses before the compass hits the ground?

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Rojas, G.

October 4, 2022

Why do we have to use the quadratic equation?

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Christina Krawiec

Rutgers, The State University of New Jersey

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Video Transcript

in this problem. The hot air balloon is moving up at a velocity of 2.5 meters per second and a balloonist in the balloon and the hot air balloon has dropped a compass. The question asks us to find if the hot air balloon at that point is three meters above the ground. How long will it take the compass to hit the ground? So we know our are given initial velocity is 2.5. The distance that the compass isn't a fall is three meters, cause that's where it starts and it hits the ground below. But it's in the negative direction. It's going down where the initial velocity is going up, so that needs a positive sign. The acceleration is the acceleration due to gravity which points down always. That's negative 9.8 meters per second square. What we're looking for is T. What we're looking for is T so we can set up our equation using the expression Why is equal toe be not tea plus 1/2 80 squared. Why is negative? Three initial velocity is 2.5 meters per second. We don't know t plus 1/2 negative 9.80 meters per second squared Kang's t squared. So we have a t n a t square toe. Looks like we're going to have to use the quadratic equation. So I'm gonna move everything to the left hand side, um so that I can set the expression up to equal zero. So living my t square term over to left, it becomes positive. Half of 9.80 is 4.90 So 4.90 meters per second squared comes T square moving my T term. The one with the 2.5 in front of it to the left will make it negative. So negative 2.5 meters per second times t, and then this one is already on the left hand side. So it stays negative. That's all equal to zero. So we'll change color here. This term is my A. This term is be and this term is seek. My quadratic equation states that your variable in this case T is equal to negative B plus or minus the square root a B squared minus four a c all divided by two a. So we could plug our values in and find an answer. 40 So T is equal to negative. B and B is negative 2.5. So I have to remember to have the negative there. I could drop my unit for this part. So it's behaved a little messy, plus or minus the square root of negative 2.5 squared. Which, of course, when you square the negative pole go away minus four. 4.9 with my A and C is negative. Three. Make my square root a little bit longer there, and that's that's all divided by two A and A's 4.9. So let's simplify what's inside the radical 1st 2.5 squared minus 4 10 4.9 times negative. Three. You put that in your calculator and take the square root. It ends up being How does put the plus or minus sign here? 8.65 So double check your mouth there, but make sure this becomes positive. 2.5 negative 2.5 When you square at the negatives goes away. You have two negatives here, so the user basically positives. Take the square root. You get 8.65 and then negative times. Negative 2.5 is positive to plant times. And then I'm gonna divide by two times 4.90 which is 9.8. So I have two solutions here. This is a quadratic equation. Um, one of them is going to give the a negative answer. If I do 2.5 minus eight, you're gonna get negative five point something. So that doesn't make sense when it's time. Because this whole point is to find the time so we can ignore the negative solution. We have 2.5 plus 8.8 point 06 5/9 60.8, which would give you 10.56 over 9.80 And that will give you a time off 1.7 seconds. So that is the answer for how long it takes the compass to hit the ground.

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Problem

A ball is thrown straight upward and rises to a m…

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Watch More Solved Questions in Chapter 2

Video Transcript

John D. Cutnell, Kenneth W. Johnson

Physics

Christina Krawiec

Liev Birman

Zachary Mitchell

Aspen Fenzl

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Physics

Christina Krawiec

Liev Birman

Zachary Mitchell

Aspen Fenzl

John D. Cutnell, Kenneth W. JohnsonPhysics

Christina Krawiec

Liev Birman

Zachary Mitchell

Aspen Fenzl

John D. Cutnell, Kenneth W. Johnson

Physics