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.

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Divide the interval [34, 64] into n = 5 subintervals: [34, 40], [40, 46], [46, 52], [52, 58], and [58, 64]. Note that the width of each subinterval is Δt = 6 seconds.

For the left Riemann sum, use the velocity values at the left endpoints of each subinterval: v(34), v(40), v(46), v(52), and v(58). Multiply each velocity value by the subinterval width Δt to approximate the distance traveled in each subinterval.

For the right Riemann sum, use the velocity values at the right endpoints of each subinterval: v(40), v(46), v(52), v(58), and v(64). Multiply each velocity value by the subinterval width Δt to approximate the distance traveled in each subinterval.

Sum the results of the left Riemann sum and the right Riemann sum separately to estimate the total distance Felix fell while traveling at supersonic speed.

Compare the left and right Riemann sums to understand the range of possible distances traveled during the interval [34, 64].

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주요 개념

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Riemann Sums

Riemann sums are a method for approximating the total area under a curve by dividing it into smaller subintervals. In this context, the left and right Riemann sums use the function values at the left and right endpoints of each subinterval to estimate the area, which corresponds to the distance fallen by Felix during his supersonic speed. This technique is fundamental in calculus for understanding integration.

Velocity and Distance Relationship

The relationship between velocity and distance is crucial in understanding motion. Velocity, defined as the rate of change of position with respect to time, can be integrated over a time interval to find the total distance traveled. In this problem, the area under the velocity-time graph during the specified interval represents the distance Felix fell while traveling at supersonic speed.

Supersonic Speed

Supersonic speed refers to speeds that exceed the speed of sound, which is approximately 343 m/s at sea level. In the context of Felix's fall, understanding supersonic speed is essential as it indicates the phase of his fall where he traveled faster than sound. This concept is important for interpreting the velocity graph and calculating the distance fallen during that specific time frame.