Question

If a body of mass m falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity t sec into dv the fall satisfies the differential equation m- mg-kv, where k is a constant that depends on the body's aerodynamic properties and the density of the air. (Assume dt that the fall is short enough so that the variation in the air's density will not affect the outcome significantly.) Complete parts (a) through (c) below gk mg tanh k a. Show that y satisfies the differential equation and the initial condition that v 0 when t 0. m dy for y dt mg tanh ki gk First, find the left side of the differential equation, m- dv m dt gk mg Now, find the right side of the differential equation, mg-kv for v Choose the correct answer below tanh k ce mc tanh gk OA mg-k v2 mg-k m ib gk O B. mg- mg-mg tanh tie gk O C. mg-kvmg tanh 2 Click to select your answer(s). E E E Which hyperbolic identity demonstrates that the expressions from the two previous steps are equal to each other? O A. coth 2(x)-1-csch2(x) O B. sinh (2x) 2 sinh x cosh x O C. sech 2(x)= 1- tanh2(x) 2()-sin2(x)- 1 O D. cosh Which of the following are used in showing that v 0 when t 0? Select all that apply. B. sech "x= tanh x+ 1 A. 0 0 mg k D. mg 0 0 k C. tanh (0)=1 F. tanh (0)0 E. kg mg =g m k H. gk 0 is undefined G. gk 0 0 m Click to select your answer(s). b. Find the body's limiting velocity, lim v. to0 The body's limiting velocity is c. For a 170-lb skydiver (mg 170), with time in seconds and distance in feet, a typical value for k is 0.006. What is the diver's limiting velocity? The diver's limiting velocity is approximately (Round to two decimal places as needed.) Click to select your answer(s).

Answer 1

Ansuer difkennul quution: mdv -mj-ky dt (q) (2) yubt Tu k LHS ( = m d clt clt TYn K ()] mm d tun h + dt dt k se'hz) Seh*z as He knnu el tan bi * (EE mi Sech dv m dt RHS. -k - mf tunh k (--- tunh Kł ayunt-T m Sec h 1-tanht

ак + mj (4) m hich hyperbolil idtnity demos trutt thut the txpn1m the tuuo pMtvlouJ ^Hpi еqчal to Cach oher at Hc hurt m dv (E*) - m mSech k dt RHS Ky Jk + Tm dv dt as Hs JKSech2 tyi |1k mla2 Stch Sech 4.H.S = R.Hs 1-tantx Scchx OpHm Cc) is (orrect

Jhin t= O, V-D as cxpMcsism of V k (m mtan h(o) - tanhlo) mf k tan h(o) o OpHn F is Lorrect.