B Apollo Reentry Courtesy of Alar Toomre, Massachusetts Institute of Technology Each time the Apollo astronauts returned from the moon circa 1970, they took great care to reent Earth's atmosphere along a path that was only a small angle a from the horizontal. (See Figure 4 25 This was necessary in order to avoid intolerably large "g" forces during their reentry. To appreciate their grounds for concern, consider the idealized problem sp ) H/sY-3= dr up where K and H are constants and distance s is measured downrange from some reference point on the trajectory, as shown in the figure. This approximate equation pretends that the only force on the capsule during reentry is air drag. For a bluff body such as the Apollo, drag is proportional to the square of the speed and to the local atmospheric density, which falls off exponentially with height. Intuitively, one might expect that the deceleration predicted by this model would depend heavily on the constant K (which takes into account the vehicle's mass, area, etc.); but, remark- ably, for capsules entering the atmosphere (at "s = -0") with a common speed Vo, the maxi- mum deceleration turns out to be independent of K. (a) Verify this last assertion by demonstrating that this maximum deceleration is just Vỏ/ (2eH). [Hint: The independent variable t does not appear in the differential equa- tion, so it is helpful to make the substitution v = ds/dt; see Project A, part (b).] (b) Also verify that any such spacecraft at the instant when it is decelerating most fiercely will be traveling exactly with speed Vo/Ve, having by then lost almost 40% of its origi- nal velocity. (c) Using the plausible data V% = 11 km/sec and H = had to be chosen so as to inconvenience the returning travelers with no more than 10 g's. 10/(sina) km, estimate how small a Distance s Figure 4.37 Reentry path
B Apollo Reentry Courtesy of Alar Toomre, Massachusetts Institute of Technology Each time the Apollo astronauts returned from the moon circa 1970, they took great care to reent Earth's atmosphere along a path that was only a small angle a from the horizontal. (See Figure 4 25 This was necessary in order to avoid intolerably large "g" forces during their reentry. To appreciate their grounds for concern, consider the idealized problem sp ) H/sY-3= dr up where K and H are constants and distance s is measured downrange from some reference point on the trajectory, as shown in the figure. This approximate equation pretends that the only force on the capsule during reentry is air drag. For a bluff body such as the Apollo, drag is proportional to the square of the speed and to the local atmospheric density, which falls off exponentially with height. Intuitively, one might expect that the deceleration predicted by this model would depend heavily on the constant K (which takes into account the vehicle's mass, area, etc.); but, remark- ably, for capsules entering the atmosphere (at "s = -0") with a common speed Vo, the maxi- mum deceleration turns out to be independent of K. (a) Verify this last assertion by demonstrating that this maximum deceleration is just Vỏ/ (2eH). [Hint: The independent variable t does not appear in the differential equa- tion, so it is helpful to make the substitution v = ds/dt; see Project A, part (b).] (b) Also verify that any such spacecraft at the instant when it is decelerating most fiercely will be traveling exactly with speed Vo/Ve, having by then lost almost 40% of its origi- nal velocity. (c) Using the plausible data V% = 11 km/sec and H = had to be chosen so as to inconvenience the returning travelers with no more than 10 g's. 10/(sina) km, estimate how small a Distance s Figure 4.37 Reentry path
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Part A only
![Earth's along a path that was only a small angle a from the 4.37.)
Each the Apollo the took care to reenter
B Apollo Reentry
Courtesy of Alar Toomre, Massachusetts Institute of Technology
Each time the Apollo astronauts returned from the moon circa 1970, they took great care to reent
Earth's atmosphere along a path that was only a small angle a from the horizontal. (See Figure 4 37
This was necessary in order to avoid intolerably large "g" forces during their reentry.
To appreciate their grounds for concern, consider the idealized problem
- Ke/H
sp
dr²
where K and H are constants and distances is measured downrange from some reference point
on the trajectory, as shown in the figure. This approximate equation pretends that the only force
on the capsule during reentry is air drag. For a bluff body such as the Apollo, drag is proportional
to the square of the speed and to the local atmospheric density, which falls off exponentially with
height. Intuitively, one might expect that the deceleration predicted by this model would depend
heavily on the constant K (which takes into account the vehicle's mass, area, etc.); but, remark-
= – o") with a common speed Vo, the maxi-
ably, for capsules entering the atmosphere (at "“s
mum deceleration turns out to be independent of K.
%3D
(a) Verify this last assertion by demonstrating that this maximum deceleration is just
Vỏ/(2eH). [Hint: The independent variable t does not appear in the differential equa-
tion, so it is helpful to make the substitution v =
(b) Also verify that any such spacecraft at the instant when it is decelerating most fiercely
will be traveling exactly with speed Vo/Ve, having by then lost almost 40% of its origi-
nal velocity.
(c) Using the plausible data Vo = 11 km/sec and H = 10/(sina) km, estimate how small a
had to be chosen so as to inconvenience the returning travelers with no more than 10 gʻ's.
ds/dt; see Project A, part (b).]
%3D
Distance s
70
Figure 4.37 Reentry path](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00d6b501-453b-4df9-bb4e-f509d174e293%2F5633b3cd-af6a-4a72-bebd-67749a0cb55d%2Ff1rq7d_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Earth's along a path that was only a small angle a from the 4.37.)
Each the Apollo the took care to reenter
B Apollo Reentry
Courtesy of Alar Toomre, Massachusetts Institute of Technology
Each time the Apollo astronauts returned from the moon circa 1970, they took great care to reent
Earth's atmosphere along a path that was only a small angle a from the horizontal. (See Figure 4 37
This was necessary in order to avoid intolerably large "g" forces during their reentry.
To appreciate their grounds for concern, consider the idealized problem
- Ke/H
sp
dr²
where K and H are constants and distances is measured downrange from some reference point
on the trajectory, as shown in the figure. This approximate equation pretends that the only force
on the capsule during reentry is air drag. For a bluff body such as the Apollo, drag is proportional
to the square of the speed and to the local atmospheric density, which falls off exponentially with
height. Intuitively, one might expect that the deceleration predicted by this model would depend
heavily on the constant K (which takes into account the vehicle's mass, area, etc.); but, remark-
= – o") with a common speed Vo, the maxi-
ably, for capsules entering the atmosphere (at "“s
mum deceleration turns out to be independent of K.
%3D
(a) Verify this last assertion by demonstrating that this maximum deceleration is just
Vỏ/(2eH). [Hint: The independent variable t does not appear in the differential equa-
tion, so it is helpful to make the substitution v =
(b) Also verify that any such spacecraft at the instant when it is decelerating most fiercely
will be traveling exactly with speed Vo/Ve, having by then lost almost 40% of its origi-
nal velocity.
(c) Using the plausible data Vo = 11 km/sec and H = 10/(sina) km, estimate how small a
had to be chosen so as to inconvenience the returning travelers with no more than 10 gʻ's.
ds/dt; see Project A, part (b).]
%3D
Distance s
70
Figure 4.37 Reentry path
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