While on vacation, a tourist walked with a velocity of 1.25 m/s [275°] on the deck of a cruise ship. The
cruise ship was travelling with a velocity of 2.50 m/s [320°] on the ocean, where the water current had a
velocity of 1.00 m/s [110°].
(a) Draw an accurate vector diagram to graphically determine the resultant velocity
of the tourist with respect to an observer on the shore. Explain how to
determine the resultant velocity from the diagram.
(b) Calculate, algebraically, the magnitude of the resultant velocity of the tourist
with respect to an observer on the shore.
(c) Calculate, using trigonometry, the direction of the resultant velocity of the
tourist with respect to an observer on the shore.
(d) Determine the time required for the tourist to reach a point on the ship that is
25.0 m away at an angle of [275°]. Explain the rule used in determining the
time taken to reach a destination when solving relative motion problems.
(a) To draw the vector diagram, you would first draw a vector representing the velocity of the tourist on the cruise ship (1.25 m/s [275°]), then add the vector representing the velocity of the cruise ship (2.50 m/s [320°]), and finally add the vector representing the water current (1.00 m/s [110°]). The resultant velocity can be determined by calculating the sum of these vectors using vector addition.
(b) Using the cosine law, we can calculate the magnitude of the resultant velocity:
V^2 = (1.25)^2 + (2.50)^2 - 2(1.25)(2.50)cos(320° - 275°)
V^2 = 1.5625 + 6.25 - 5cos(45°)
V^2 = 7.8125 - 5(0.7071)
V = √4.3183
V = 2.08 m/s
(c) Using trigonometry, we can calculate the angle of the resultant velocity:
tanθ = (1.25sin275° + 2.50sin320°) / (1.25cos275° + 2.50cos320°)
tanθ = (1.0622 - 1.7064) / (0.7849 + 2.1511)
tanθ = -0.6442 / 2.9360
θ = tan^(-1)(-0.2189)
θ ≈ -12.4°
(d) The time required for the tourist to reach a point on the ship that is 25.0 m away at an angle of [275°] can be calculated using the formula:
time = distance / velocity
time = 25.0 / 1.25
time = 20 seconds
The rule used in determining the time taken to reach a destination when solving relative motion problems is to calculate the distance between the starting point and the destination, and then divide it by the relative velocity.