Question

Consider two ships in the lake, as shown in the picture. Ship A moves at a speed u parallel to the x axis. Ship B moves at a speed of 2u along the y-axis. At the time t = 0, the ship B and the ship A are at a distance a from each other and they have the same coordinates y. a) Find the relative speed of the ship B relative to the ship A and the direction of this speed. (2 points) b) How long will the distance between the ships be equal to 3a? (2 points) c) Find the shortest distance between the ships and at what point in time this will happen. (2 points) e) Suppose that at t = 0 the speed of the ship A remains equal to u, but its direction can be adjusted. How ship A should be directed so that the minimum distance between the two ships is minimal. How much is this minimum shorter than the result obtained in part (c)? (4 points)

2и а B и

Answer 1

Zuj-u Solution a) Relative velocity of ship B rela to the Ship A = V = vid .. Relative speed of ship B to the ship A = 140°+U? V5 U solution b) na nola S= zabele Cľ SVAD y 20 NB. y a>> n A Va+y² = 30 n+ y² = gaz =>(ut-a) 7 40t' = 90 x = ut -a {y=aut

root vett a² 2 2 2 - Quta, + 40²t² = ga 5u²t² – Ruta 4a²-ga 1 - 2 0 4 + 40*- 45 )(aº-guºn 100² t = Qua I UW 180 1²-16 2² = 2at Visou 160² lou. I lou t= a + Asu²-40² 180116d all entlook 5U autt t = v 40°a? -4(502) 680) = 2uth + 2 v4r ua 10u² to u2 t = allt vil 50 . Time time by ship after which the distance between the ship = 3a is a (1+141) 5U = Solutionc) Let the shortest distance between the Ships - S. :: S = (a -ut) + 4 ut? = 200-U+)EU) + suit =(-atut) u + 188² Va-ut² + 40²72 > 80²7 = aut u²t & Tut=au It = 28 ds dt = 0 ds dt 2 70

. • the distance between after t = a time 7U A and B is shortest and the shortest distance between them is s=1 40 la va-ajº+ + 492 72 = 2a via 7 i. s= ax2 110 7 solution e) Perpendicular distance between two objects is the shortest tren y 0 0 s- U 12U in Usoola

s = acoso tand- 20- Usino - U coso Cot o -- 2ut using UCOS O - cos²e - (sino-2) sino cos?o= sino-asino Vi-sino = sino (sin 0-2) & (sin?a-osinas = 1-sin? sinto - 4 sino + 4 sino = 1-sina sinto - 3sino +45 insan1=0. sino = 0.8796 → A = 61.59 s'= acos (61.599) - 0.47 57 A S-S! = 0.ga- 094757 a 0.427 a. s' is 0.427a Shorter than s at 0 = 61.59°, the minimum distance between the two ships is minimal