Question

Two boats are heading away from shore. Boat 1 heads due north at a speed of 2.81 m/s relative to the shore. Relative to Boat 1, Boat 2 is moving 31.5° north of east at a speed of 1.50 m/s. A passenger on Boat 2 walks due east across the deck at a speed of 1.29 m/s relative to Boat 2. What is the speed of the passenger relative to the shore?

Answer 1

Let
vs1 = velocity of boat 1 relative to shore
v12 = velocity of boat 2 relative to boat 1
v2p = velocity of passenger relative to boat 2,
vsp = velocity of passenger relative to shore
vs2 = velocity of boat 2 relative to shore

east = +ve x direction,
north = +ve y direction
i = unit vector along x axis,
j = unit vector along y axis,

vs1 = 2.81 m/s j--------------------------------(1)
v12 = 1.5 m/s(cos 31.5 deg i + sin 31.5 deg j)
Or v12 = 1.5 m/s * cos 31.5 deg i + 1.5 m/s * sin 31.5 deg j
Or v12 = 1.28 m/s i + 0.78 m/s j------------------------(2)

v2p = 1.29 m/s i--------------------------(3)

vsp = vs2 + v2p---------------------(4)
vs2 = vs1 + v12---------------------(5)

Using (5) in (4),
vsp = vs1 + v12 + v2p
Using (1), (2), (3) in the above,
vsp = 2.81 m/s j + 1.28 m/s i + 0.78 m/s j + 1.29 m/s i
= (1.28 + 1.29)m/s i + (2.81 + 0.78)m/s j
= 2.57 m/s i + 3.59 m/s j
Speed of passenger relative to shore = magnitude of vsp
= sqrt(2.57^2 + 3.59^2) m/s
= 4.415 m/s

Ans: 5.6 m/s