Free fall On October 14, 2012, Felix Baumgartner stepped off a...

Question

Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earth’s surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)

(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints x₀ = 34 , x₁ = 40 , x₂ = 46 , x₃ = 52 , x₄ = 58 , and x₅ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.

Accepted Answer

Hello. In this video, we are told that a skydiver jumps from a plane, and her velocity in meters per second is recorded at various times during her descent. Between T is equal to 20 seconds and T is equal to 50 seconds, she is in free fall. We want to divide the interval from 20 to 50 into 5 subintervals with grid points at 2026, 32, 38, 44, and 50. The recorded velocities at 20 is 120, 135, 140, 130, 125, and 115. And we want to use the left Ryman sum to estimate the distance she fell during the interval. So, let's go ahead and just list out the given information. First, we are working with the time interval between 20 and 50, and we want to divide this time interval into 5 sub-intervals. So, let's go ahead and draw a number line between 20 and 50. Now, we are told that we are splitting this up into 5 sub-intervals. And we are given the endpoints of these intervals. These intervals were taken at the various time points, and those time points were given to us as 2026, 32, 38, 44, and 50. We are even given the recorded velocities for each of these time points. So, in order to figure out the distance that she fell during this interval, we want to go ahead and take the sum of the 5 sub-intervals using the left Riemann sums. Now, the first thing we want to do is you want to go ahead and determine the width of our time intervals. That is going to be determined as delta T. Delta T is defined as our upper boundary minus our lower boundary, divided by the amount of intervals that we have. Now, our upper value is going to be 50, so B is 50, and we are going to subtract our lower boundary, which is 20. And since we are dividing this by 5 some intervals, we are going to divide this distance by 5. This is going to simplify to be 30 divided by 5, which will further simplify to be 6. So we know that the width of time between each interval is 6 seconds. Now, let's go ahead and write out the given sum or the summation for the total amount of distance that she fell. That summation is going to be defined as the summation from I is equal to 1 to 5 of the velocity function of X of I multiplied by the width of our time intervals. This can further expand to be delta T multiplied by the quantity. Of V of X1. Plus V of X 2 plus V of X 3. Plus V of X 4 plus V of X 5. Now, in this case here, V of X1, X2, X3, 4, and X 5 are going to be the left Ryman sum endpoints that we want to plug in for the velocity function. So, the 1st 5 left endpoints of our time intervals are going to be all the values between 20 and 44. In fact, we're even given those values in the problem. So, if we were to plug in those values and the defined delta T that we sought for, we can rewrite the summation to be 6. Multiplied by the value of the velocity at 20, which is 120, plus the velocity at 26, which is 135, plus the velocity at 32, which is going to be 140, plus the velocity at 38, which will be 130. Plus the velocity at 44, which is going to be 125. Now, the sum of all these values together is going to be 650. So we are left with 6 multiplied by 650, which is going to further simplify to be 3900 m. So the total distance that she fell from 20 to 50. With the respective time intervals is going to be 3900 m, and with that being said, the solution to this problem is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Key Concepts

Riemann Sums

Velocity and Distance Relationship

Supersonic Speed

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