Question
1.) Consider a space probe that is traveling through an interstellar cloud (far from any large gravitational masses). The probe has constant thrust 𝐹𝑡ℎ𝑟𝑟𝑟𝑟 ≡ 𝑚̇𝑢Thu and experiences a linear drag force 𝐹𝐷 = −𝑏𝑏
(a) The probe runs out of fuel when the ratio of its final mass to initial mass is ɳ ≡ 𝑚𝑓/𝑚0. What is the speed of the probe when it runs out of fuel? Hint: Use the chain rule to rewrite the rocket equation for velocity as a function of mass instead of time: 𝑑𝑑/𝑑𝑑 = (𝑑𝑑/𝑑𝑑)𝑚̇ . Your final answer for v should have ɳ to some power.
(b) Derive an approximate expression for the case of very small drag in the form of a series of ln ɳ. Hint: Rewrite ɳ = exp(ln ɳ). That In ɳ as your variable and use the Taylor series expansion for the exponential.
You only need to work with the first couple terms. If you take the limit of b→ 0, is the result consistent with the case of zero drag?

2.) As a raindrop falls through a cloud, it collides with smaller droplets of mist and grows in mass with a rate proportional to the cross-sectional area of the drop and to the speed: 𝑑𝑑/𝑑𝑑 = 𝐾𝑟²𝑣. It turns out that the drop falls with constant acceleration 𝑎 < 𝑔. Find the acceleration a. Hint: Do NOT just differentiate 𝑑(𝑚𝑚)/𝑑𝑑, but start with the impulse-momentum theorem in differential form, like we did in the derivation of the rocket equation. Your "system" should include the raindrop itself and a small mass ∆𝑚 of droplets with which the raindrop will collide in time ∆𝑡. Consider the total momentum before and immediately after the collision. After you take a limit ∆𝑡 → 0, this will yield a nonlinear differential equation for mass and velocity of the raindrop. (It turns out, you get the same result as if you did just differentiate the momentum, but that is NOT the proper way of doing it. It gets the "right" result here, but as discussed in class, it would give you the wrong result for the rocket equation.) In order to solve for the mass and velocity, you will also need an equation that relates the mass and radius of the raindrop (assume constant density water): 𝑚 = 𝜌water4𝜋𝑟³/3. The system of differential equations is really messy to solve. To make the math a little easier, you can assume from the beginning that 𝑣 = 𝑎𝑎 with a being constant.

3.) An inverted garbage can of weight W = 30 N is suspended in air by water from a geyser. The water shoots up from the ground with an initial speed 𝑣0= 20 m/s at a constant rate 𝑚̇ = 2.0 kg/s. Once equilibrium is established and the can is being suspended in steady-state, what is the height of the garbage can assuming that the water bounces elastically. Hint: the velocity of the water in the stream is not constant in height, but slows down because of gravity. You should be able to derive velocity as a function of height z from kinematics. Also, λ is NOT constant along the stream, but the product of 𝜆𝑣 is constant. This is conservation of mass for the stream: if you consider any small segment of the stream, the rate of volume flowing into the segment is 𝜆1𝑣1, and the rate flowing out of the segment is 𝜆2𝑣2. Assuming that no water piles up in the segment, these two volume flow rates must be equal.
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        Backtrace
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        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        1.65mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        1.07mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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        Bindings
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        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
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        • 18. app/Repositories/HomeworkLibraryRepository.php:39
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      • select * from `subjects` where `subjects`.`id` in (401)
        1.32mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
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      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 18210 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.98mstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
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        1.89mstwenty4_siteHomeworkLibrary.php#260
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
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        780μstwenty4_siteHomeworkLibrary.php#260
        Bindings
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
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