I need this homework done
© 2022 Y. Nievergelt Problem on Integrals 1 Problem 1 Denote by q (t) is the position at time t of an object moving on a line. The function q need not be a position, q may be any differentiable function. The derivative, q̇, of q need not be a velocity, q̇ is just the derivative of q. Assume that q is differentiable, so that its derivative, q̇ = q′, is the velocity, v: v (t) = q̇ (t) = q′ (t) (1) Position and velocity are given, or measured, at three times, t0, t1, and t2, with t0 < t1="">< t2="" (2)="" v="" (t0)="v0"> v1 = v (t1) > v2 = v (t2) ≥ 0 (3) 0 ≤ q (t0) = q0 < q1="q" (t1)="">< q2 = q (t2) (4) assume also that the position increases as the velocity decreases on the interval of time [t0, t2]. the problem is to establish a procedure to estimate a time t∗ between t0 and t1 when the position q∗ = q (t∗) is exactly halfway between q0 and q1: q∗ = q (t∗) = q0 + q1 2 (5) assume that v is affine: for some constant a, v (t) = v0 + (t− t0) · a (6) (1.1) calculate a in terms of t0, t1, v0 and v1. (1.2) find a formula for q (t) for every t in terms of t, t0, t1, q0, q1, v0, v1. (1.3) then solve equation (5) for t∗ in terms of t0, t1, q0, q1, v0, and v1. (1.4) calculate the velocity when the position is halfway, v (t∗) © 2022 y. nievergelt problem on integrals 2 potentially useless background numerical values are in reference [2] in sex- agesimal notation (base sixty). time is in days, velocity is in degrees per day. t0 = 0 t1 = 1, 0 t2 = 2, 0 [days] = sixty = twice sixty v0 = 0; 12 v1 = 0; 9, 30 v2 = 0; 1, 30 [ ◦/day] = twelve sixtieths = nine and a half sixtieths = one and a half sixtieths reference [2] is available from the library. supplementary materials are on line. the book [1] presents more context but does not contain anything about this computation because it was written before the discovery published in [2]. references [1] mathieu ossendrijver. babylonian mathematical astronomy: procedure texts. sources and studies in the history of mathematics and physical sciences. springer, new york, 2012. [2] mathieu ossendrijver. ancient babylonian astronomers calculated jupiter’s position from the area under a time-velocity graph. science, 351(6272):482–484, 29 jan- uary 2016. http://science.sciencemag.org/content/351/6272/482. full, supplementary material at http://science.sciencemag.org/content/ 351/6272/482/suppl/dc1. q2="q" (t2)="" (4)="" assume="" also="" that="" the="" position="" increases="" as="" the="" velocity="" decreases="" on="" the="" interval="" of="" time="" [t0,="" t2].="" the="" problem="" is="" to="" establish="" a="" procedure="" to="" estimate="" a="" time="" t∗="" between="" t0="" and="" t1="" when="" the="" position="" q∗="q" (t∗)="" is="" exactly="" halfway="" between="" q0="" and="" q1:="" q∗="q" (t∗)="q0" +="" q1="" 2="" (5)="" assume="" that="" v="" is="" affine:="" for="" some="" constant="" a,="" v="" (t)="v0" +="" (t−="" t0)="" ·="" a="" (6)="" (1.1)="" calculate="" a="" in="" terms="" of="" t0,="" t1,="" v0="" and="" v1.="" (1.2)="" find="" a="" formula="" for="" q="" (t)="" for="" every="" t="" in="" terms="" of="" t,="" t0,="" t1,="" q0,="" q1,="" v0,="" v1.="" (1.3)="" then="" solve="" equation="" (5)="" for="" t∗="" in="" terms="" of="" t0,="" t1,="" q0,="" q1,="" v0,="" and="" v1.="" (1.4)="" calculate="" the="" velocity="" when="" the="" position="" is="" halfway,="" v="" (t∗)="" ©="" 2022="" y.="" nievergelt="" problem="" on="" integrals="" 2="" potentially="" useless="" background="" numerical="" values="" are="" in="" reference="" [2]="" in="" sex-="" agesimal="" notation="" (base="" sixty).="" time="" is="" in="" days,="" velocity="" is="" in="" degrees="" per="" day.="" t0="0" t1="1," 0="" t2="2," 0="" [days]="sixty" =="" twice="" sixty="" v0="0;" 12="" v1="0;" 9,="" 30="" v2="0;" 1,="" 30="" [="" ◦/day]="twelve" sixtieths="nine" and="" a="" half="" sixtieths="one" and="" a="" half="" sixtieths="" reference="" [2]="" is="" available="" from="" the="" library.="" supplementary="" materials="" are="" on="" line.="" the="" book="" [1]="" presents="" more="" context="" but="" does="" not="" contain="" anything="" about="" this="" computation="" because="" it="" was="" written="" before="" the="" discovery="" published="" in="" [2].="" references="" [1]="" mathieu="" ossendrijver.="" babylonian="" mathematical="" astronomy:="" procedure="" texts.="" sources="" and="" studies="" in="" the="" history="" of="" mathematics="" and="" physical="" sciences.="" springer,="" new="" york,="" 2012.="" [2]="" mathieu="" ossendrijver.="" ancient="" babylonian="" astronomers="" calculated="" jupiter’s="" position="" from="" the="" area="" under="" a="" time-velocity="" graph.="" science,="" 351(6272):482–484,="" 29="" jan-="" uary="" 2016.="" http://science.sciencemag.org/content/351/6272/482.="" full,="" supplementary="" material="" at="" http://science.sciencemag.org/content/="">