TIInterActive FileA Tale of two cheeses - Adrian Oldknow a_oldknow@compuserve.com Chapter 1: The flight of the Cheddar - (expurgated version) The acceleration of the Cheddar is that of gravity - i.e. g = 9.8 m/s/s approximately (as observed by Galileo one sunny afternoon in Pisa). Assuming that the balloon was stationary and that the cheese left the balloon at time t = 0 with downward velocity v = 0 then make up a table of its velocity v m/s after times t =1, 2, 3, .. . , s. t 0 1 2 3 4 5 v 0 9.8 What would a graph of v against t look like? If the speed of sound in air (Mach 1) is about 330 m/s how long will it be before the Cheddar breaks the sound barrier? Is this likely? Give some reasons. The differential equation is just  (a constant), or v' = g, with v = 0 when t = 0. The difference equation  is just that of an arithmetic progression (additive sequence), where: .  . The solution of the difference equation is:  . The anti-derivative of the differential equation is v = gt + c , and. from the initial condition, c = 0 so v = gt is the solution. If we integrate the differential equation we have:  with the same solution, and this is the area under the graph of v' = g . We can model the number sequence using the SEQ plotting Mode of a TI graphic calculator:    An alternative is to use a spreadsheet - either Excel, or the spreadsheet of TI InterActive! (TII!) Here we can compare the scatterplot of the v sequence against t, and the graph of v = gt  An alternative (usually preferable) approach is to use the statistical lists of the calculator or TII!     Now we can plot a scattergram of vl against tl and fit the function v = gt .  The area under graph v = g from t = 0 to t gives the change in velocity i,e, v = gt. The time to reach the sound barrier would be:  sec. This seems very long - but we need to have some idea of the maximum height of an ordinary balloon - perhaps 1000m?? See Adventure balloons .ix YP)PC=P:PPPbPcPP*PPPPPPPDPP,PEPF POPPP P PPP1P2P6P8P>P? 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1000000 0 74000000 1801c 11d0000 ffff0000 ffff ffff0000 1 1 0 0 0 1f0000 8020 0 0 0 10000 1 0 0 801c29 0 0 0 ff000000 1ff 0 0 0 0 1f000000 0 6b006600 6700 0 0 1000000 1000000 1c000000 180 2c100 ffffff00 ff 1ffff00 1000000 0 0 0 0 2a001f00 2a000000 2a000000 2a000000 0 33000000 c1000006 ff000002 ffffffff ffffffff 1ff 100 0 0 0 100 0 300 4d430500 2545345 800000 0 30306431O41b 32 32 1 1 1 0 ffff0007 b0002 796c4743 74536870 10656c79 ffffff 0 0 90000000 1 7010000 54000000 73656d69 77654e20 6d6f5220 6e61 0 0 0 ff000000 ff000000 f000ffff 64000000 8000000 6e617453 64726164 0 1 80010006 0 0 0 0 190 1000000 7 626d7953 6c6f 0 0 0 0 0 0 ff ffffff f0 64 64745309 626d7953 16c6f 10000 0 8001 0 0 0 2bc0000 0 70100 79530000 6c6f626d 0 0 0 0 0 0 ff0000 ffff0000 f000ff 640000 530d0000 79536474 6c6f626d 646c6f42 1 1 80010000 0 0 0 0 190 1000001 7 656d6954 654e2073 6f522077 6e616d 0 0 0 0 ff ffffff f0 64 64745309 6c617449 16369 10000 0 8001 0 0 0 2bc0000 10000 70100 69540000 2073656d 2077654e 616d6f52 6e 0 0 0 ff0000 ffff0000 f000ff 640000 530d0000 74496474 63696c61 646c6f42 1 1 80010000 0 0 0 0 190 1000000 7 4d204954 687461 0 0 0 0 0 0 ff ffffff f0 64 64745305 14954 10000 0 8001 0 0 0 2bc0000 0 70100 49540000 74614d20 68 0 0 0 0 0 ff0000 ffff0000 f000ff 640000 53090000 49546474 646c6f42 1 1 30000 50004 70006 1500008 1590000 e10000 0 0 0 1 0 1ffff 47430009 6870796c 606c6f43 1d000000 e1000001 0 ff000000 ff 100 0 ff000100 90001ff 6c474300 52687079 60776f 11d0000 e10000 0 ffff0000 0 1 0 ffff0001 a0001 796c4743 65546870 16d72 11d0000 ffff0000 ffff ffff0000 1 1 0 0 0 ffff0000 140001 796c4743 69536870 656c706d 706d6f43 7265736f 0 0 1158 ffffffc4 0 0 0 0 0 0 0 0 0 1ffff 10000 0 0 0 10000 0 30000 43050000 5453454d 80000002 20de02 0(33 10001 0 0 0 0 13601 0 10000 0 0 0 0 02@@6integral((1),(v),(0),(v)) = integral((g),(t),(0),(t))Times New RomanTimes New Roman      0j ==.Times New Romanww 0w#f>- --Symbol[# Tww 0w#f>- -- 2  2 V 2 =V 2 BB 2 . 2 . 2 bTimes New Romanww 0w#f>- --!v!0g- --!1 ! TI Math# Tww 0w#f>- --!- --!vd! != ! - -- 2 | 2 |V 2 =|V 2 B|B 2 .| 2 .| 2 |- --!t|!04- --!g! #- --!_- --!t''NANIt + c , and. from the initial condition, c = 0 so v = gt is the solution. If we integrate the differential equation we have: with the same solution, and this is the area under the graph of v' = g .0-<P=:PwPxPP P<P=*PgPhPyP{PP>nl*@dd ;7 CTxobjItem'ࡱ>   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry$`VF "3;Contents8OlePres0008w R KQlt.8BM.86(7                                                                                                                                                                                                                                                         !"#$%&'()*+,-./0123456789:;<=>?6U M 7,.'( xw ~8  w ~8 ?  C (7                                                                                                                                                                                                                                                   'NANIDvl*@dd ;7 CTxobjItem(ࡱ>   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry$`VF "3;Contents8OlePres0008  w R KQlt.8BM.86(7                                                                                                                                                                                                                                     !"#$%&'()*+,-./0123456789:;<=>?6U M 7.'(    w ~8 w ~8 ?  C (7                                                                                                                                                                                                                                 'NANIMDl*@dd ;7 CTxobjItem)ࡱ>   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root Entry$`VF "3;Contents8OlePres0008  w R KQlt.8BM.86(7                                                                                                            !"#$%&'()*+,-./0123456789:;<=>?6U M 7.'( w ~8 w ~8 ?  C (7