Please answer all part of this question, even pt.d with Mathematica This is the

Question

Please answer all part of this question, even pt.d with Mathematica

This is the page needed in pt (a):

1. There is a common myth that a penny dropped from the Empire State building can kill a person walking on the sidewalk below. Let's do our own investigation! (a) (1 point) Look up the size and mass of a penny. Using the drag coefficients given by Taylor (pp 44-45), which means we are assuming a spherical penny , find the terminal speed of a dropped penny, taking into account both linear and quadratic terms together. (b) (1 point) Write down the differential equation for the motion of a falling penny. Keep both linear and quadratic terms. Clearly articulate your sign conventions. (c) (2 points) For what range of speeds will (i) the linear term of air resistance dominate over the quadratic term? (ii) the quadratic term dominate over the linear? Discuss the relative importance of linear and quadratic drag here - if you had to pick just ONE term (linear or quadratic), which would you use, and why? (d) (3 points) If we keep both linear and quadratic terms, we have a non-linear differ- ential equation which we can only solve numerically. Use your favorite numerical differential equation solver to determine and plot velocity and position of the penny as a function of time, as it falls from the top of the Empire State building (381 m). Also find the time it hits the ground, and the speed with which it hits. Include your code and plots with your homework. Mathematica users might find this screencast on using NDSolve (Mathemat- ica's built-in numerical differential equations solver) helpful: http://youtu.be/ zK06v0wOKdI Be Elegant: You could use Mathematica's EventLocator method to stop the integration when the penny hits the ground. Helpful link for that: http://goo gl/Mkz3i For this part of the problem, print out the relevant section of your code and the plots along with any explanation needed. (e) (1 point) Compare the result for "final velocity" from your numerical results (part d) with what you got in part a, and also what you get by assuming JUST the one dominant drag term you chose in part b. Comment. (f) Bonus: (2 points extra credit): Could this penny kill someone? Use basic freshman physics and make crude (but quantitative) estimates to discuss whether vou think the myth is true or not

Explanation / Answer

a) given f(v) = bv+cv^2

The velocity of a freely falling body is given by V= U+at

   U is initial velocity =0 for free falling

   a is acceleration due to gravity g =9.8 m/s^2

t is time

   v=at=9.8t

   t=v/9.8

equation of motion of penny ,

Force due to gravity - drag force =mass of penny ( acceleration of penny    acceleration of penny =dv/dt

force due to gravity =mg

   drag force =bv+cv^2

mg- (bv+cv^2) =mdv/dt

   (mg-(bv+cv^2)) dt =mdv

( mg-(bv+cv^2))t =mv

   ( mg-(bv+cv^2))v/9.8 =mv

cv^2+bv= 0

   v =-b/c =- beta D/gamma D^2=- beta /gamme D

   terminal velocity v= (1.6*10^-4)/0.25 D

v =(6.4*10^-4)/D m/s D is diameter of penny

c) for velocity zero to half of terminal velocity linear term of air resistance dominate over quadratic term. later quadratic term takes over.

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