MAST10005 Calculus 1 3. Suppose that the positions of two particles called Chenyan and John at the time t are given by the parametric curves: r1(t) = 2 cos(t) sin?(t)i + sin(t) j r2(t) = sin(t)i + cos(t)j (a) Express the set X = {te R : y1(t) = y2(t)} in abbreviated form. Your answer to Web Work Problem 3c should be useful. Note that X is infinite. Stilt ) = 2cosct > sin = (+ ) it since>j As we can tell from the graph. (12 (t ) = sinctsi + cosct ) j t = 4 + KIT for some k E Z yi(t ) = sinct ) yo (t ) = cosct ) To Let yilt ) = yz( ) = > sinct ) = cosct ) Thus , gen Sin(t ) Cosct coset ) cos(t > = tan ( t ) = 1 X = $ 4 + KTT / KER]. (b)) Use your answer to (a) to find the set C of t-values for which there are collisions between the particles. collision occur at t-values at which rict ) = act ? Thus , Xilt ) = 42 (t ) and 4,(+ ) = 4 (t ) 2 Coscto Sin ? ( t ) - Sin ( t ) = O t = 4 +KIT for some KE Z Since) ( 2 coset ) sin(t ) -1 )= 0 from double - angle formula Thingt collision occur -> since) ( sin ( 2t ) - 1 ) = 0 sin( + ) = 0 or sin ( zt ) -1 = 0 - t= KIT for some Sin(zt ) = 1 KEZ 2t = 2 + 2 TK for some KER KIT 7 t= 7 +Tik for some KEIL NMAST10005 Calculus 1 Extra space for Question 3(b), if needed: Hence C = { # + KTT / KER 3. (c) Find the two places in R2 at which the collisions you found in (a) occur. when t= # when t = hit ) = 2 cos ( # ) ( Sin ? ( # ) i + sin 14 )j vilt ) = - zit 12 = 12lt ) = =21 -5 12(t ) = Sin(4)i + cos. 14)j = Bi+ z d. two places in IR? where the collision occurs IS ( 21 2 ) , (- 2 , - 2 )MAST10005 Calculus 1 (d) Explain why the paths of ri(t) and r2(t) cross at the point j. Your answers to Problems 3a and 3b from WebWork should be useful. Is this a collision point? ERj = 30 it j = CO , 1) When t = = > hits = ofit ' j = j when t = 0 7 12 (t ) = or + lj = j. we can see rifts and ralt ) Can cross at the point j Since they are not there at the Same time, this is not a collision Point Assignment Information This assignment is worth 9% of your final MAST10005 mark. Full working should be shown in your solutions to Questions 2 and 3. There will be 1 mark overall for correct mathematical notation. Solutions will be uploaded to Canvas approximately 3 days after the deadline