ESE 318-02, Spring 2016 Homework Set #8 Due Tuesday, Mar. 22 You do not need to provide any sketches or graphs in any of these problems, even if asked for in the Zill problem statement. 1. Zill 9.1.13. 2. Zill 9.1.15. 3. Zill 9.1.19. 4. Zill 9.1.25. 5. Zill 9.1.29. Make use of Theorem 7.4.1 on pg 314 of Zill. 6. Zill 9.1.33. 7. Zill 9.1.37. 8. Zill 9.1.44. Note: there is a simplification so that the integral is easy to do. 9. Zill 9.1.45. It’s OK to assume a > 0 so |a| = a in your answers. 10. Zill 9.2.7. 11. Zill 9.2.9. 12. Zill 9.2.19. 13. Find rdtd, expressing your answer in terms of the position vector, r, and its velocity vector, v, only. 1. Zill 9.1.13. There could be a + on the y term, just like in problem 12, but the resulting curve is the same either way. (But you really have to graph it to see that.) ( )( )( )( ) ( ) ( )( )ktjtittrttyyttttztx222222222sin9sin3cos3sin9cos199cos9sin9cos19cos39cos3++==−=⇒=+=−=−== 2. Zill 9.1.15. Take limits on each term, using L’Hopital’s Rule as needed. ( ) ( )( )0,32,23220lnln322221cos22sin21155−=−→→−=−→=−=−→−→→−−−jitrtttttttttttESE 318-02, Spring 2016 3. Zill 9.1.19. ( )( )8,6,148,6,448,6,22418,3,1218,3,24,,22222222222232tteteteteetertttetteterttttertttttttttt+=+=++=ʹʹ−+=−+=ʹ−= 4. Zill 9.1.25. ( ) ( )( )tztytxkjirkttjirkjirktjttitr422242222238238331221+=+=+=++=ʹ++=ʹ++=++= 5. Zill 9.1.29. Make use of Theorem 7.4.1 on pg 314 of Zill. ( ) ( ) ( )( )[ ]( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )( ) ( )( ) ( ) ( ) ( )( )( ) ( ) ( )( )[ ]( ) ( ) ( )( )trtrtrtrtrtrdtdviiTheoremtrtrtrbaaviTheoremtrtraatrtrtrtrtrtrtrtrtrtrtrtrtrdtdtrtrtrtrdtdʹʹʹ×ʹ⋅=ʹʹ×ʹ⋅=ʹʹ×ʹ⋅ʹ⇒=×⋅=ʹʹ×ʹʹ⇒=×ʹʹ×ʹ⋅ʹ+ʹʹ×ʹʹ+ʹʹʹ×ʹ⋅=ʹʹ×ʹ⋅ʹ+ʹʹ×ʹ⋅=ʹʹ×ʹ⋅1.4.7001.4.700∵∵ 6. Zill 9.1.33. ( )kjiktjtitdttjtti159432321432212132++=++=++−−∫ 7. Zill 9.1.37. ( ) ( )( ) ( )( ) ( )ktjtittrkjiCrCktjttitr1231620363232++−++=+−==+++= 8. Zill 9.1.44. Note: there is a simplification so that the integral is easy to do. ( )( )( )( )( )( )[ ]3113210332102102224233223332323241292,32,3,3,3=+=+=+=ʹ+=+=++=ʹ=ʹ=∫∫ttdttdttrtttttrtttrttttrESE 318-02, Spring 2016 9. Zill 9.1.45. It’s OK to assume a > 0 so |a| = a in your answers. ( ) ( )( )( )( ) ( )1cossincossinsincoscossincossin22002222=+=ʹ⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎠⎞⎜⎝⎛−=ʹ⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛===ʹ==+=ʹ+−=ʹ∫∫srjasaaiasaasrjasaiasasrtaduaduursatatatrtjatiatrtt 10. Zill 9.2.7. ( )( ) ( )( ) ( )( )1432116216232132222232=++=ʹ=+=→+=ʹʹ++=→++=ʹ++=rspeedkjatkjtrkjivkttjitrktjttitr 11. Zill 9.2.9. It passes through x-y plane when z is zero. ( )( ) ( )( )( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )kjirakjirvkirakjrvtortttktjitrktjttitrkttjttittr2302555731055220052005005262522325222232++=ʹʹ=++=ʹ=+=ʹʹ=−−=ʹ===⇒=−++=ʹʹ−+−+=ʹ−+−+= 12. Zill 9.2.19. ( )( ) ( ) ( )( ) ( ) ( )rvrtaatjrtirtrtartvvtjrtirtrtvtjrtirtr202020200000sincoscossinsincos===⇒−−=ʹʹ===⇒+−=ʹ=+=ωωωωωωωωωωωωESE 318-02, Spring 2016 13. Find rdtd, expressing your answer in terms of the position vector, r, and its velocity, v, only. ( ) ( ) ( )( ) ( ) ( ) ( )( )vcomprrvrrrvrvrrrrrrrrrrdtdrrrrdtdrdtdr=⋅=⋅⋅=⋅⋅=⋅ʹ+ʹ⋅⋅=⋅⋅=⋅=−−−2/12/1212/1212/1212/12 Also, ( )( )( )(