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Ch 02: Motion Along a Straight Line
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All textbooksYoung & Freedman Calc 14th EditionCh 02: Motion Along a Straight LineProblem 2

Chapter 2, Problem 2

A rocket starts from rest and moves upward from the surface of the earth. For the first 10.0 s of its motion, the vertical acceleration of the rocket is given by ay = (2.80 m/s3)t, where the +y-direction is upward. (a) What is the height of the rocket above the surface of the earth at t = 10.0 s?

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Welcome back everybody. We are making observations about a hot air balloon and we are told that it initially starts off at rest but then starts rising upwards after a time or during the 1st 70 seconds. My apologies. We are given a vertical acceleration as a function of time equivalent to 0.8 m per second cube times T. And we are tasked with finding what the height of the hot air balloon is. After 70 seconds. In order to figure this out. We are going to need to know what our height is as a function of time and here's how we are going to do this. We know that our height is equal to the integral of zero to T. Of our velocity as a function of time D. T. We also know that our velocity as a function of time is equal to our initial velocity plus the integral from zero to t. Of our acceleration as a function of time. So in order to find our height, we need to find the vertical velocity using our acceleration. So let's go ahead and do that. So our vertical velocity as a function of time is going to be V. V not which is zero plus the integral from zero to T. Of this equation right here. So this is 0.008 t. d.t. Giving us 0.4 t squared. Great. So then once again we are going to integrate to find our height. So our height is the integral zero to T. Of the function or equation that we just found 0.4 T square, which gives us D. T. My apologies, equals 1/7 50 T cubed. So now that we have found our height as a function of time, let's go ahead and plug in 70 seconds. Here we get 1/7 50 times cubed, which when you plug this in your calculator, we get the height at 70 seconds is 457. m corresponding to our final answer choice of B. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.
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