Relative Velocity: Solving Boat Race Circuits

[CloudKel]

Homework Statement

Relative velocity question. A power boat has cruising speed of 60 km/hr. It is
taking part in a race around three buoys arranged in an equilateral triangle. The
second buoy, B, is 5 km due East of the starting/finishing buoy, A. The third
buoy, C, is to the North of the line joining A and B. There is a current flowing at
10 km/hr from the south-west. In what direction should the boat be steered on
each leg of the race and how long will it take the boat to complete one circuit.

Homework Equations

Vb = Vb/w + Vw

The Attempt at a Solution

A to B
Vw = 5(square root 2)i + 5(square root 2)j
Vb/w 60Cos(theta)i + 60Sin(theta)j
Vb = 5(square root 2) + 60Cos(theta)i + 5(square root 2)+ 60Sin(theta)j

5(square root 2)+ 60Sin(theta)j = o
Sin(theta) =5(square root 2) / 60
theta = 6.77

5(square root 2) + 60Cos(6.77)i =66.65i
Vb = 66.65i + 0j
t = 0.075hrs
t = 270.068s

 

[Delphi51]

Welcome to PF, CloudKel!
Those answers for AB are correct. I don't see how you can do the rest unless you get more specific information about the location of C.

 

[kerimek]

B to C
Vw = 10i + 10j
Vb/w = 60Cos(theta)i + 60Sin(theta)j
Vb = 10i + 10j + 60Cos(theta)i + 60Sin(theta)j

10i + 10j + 60Sin(theta)j = 0
Sin(theta) = -10/60
theta = -9.46

10i + 10j + 60Cos(-9.46)i = 58.19i + 10j
Vb = 58.19i + 10j
t = 0.079hrs
t = 284.068s

C to A
Vw = -5i + 0j
Vb/w = 60Cos(theta)i + 60Sin(theta)j
Vb = -5i + 60Cos(theta)i + 60Sin(theta)j

-5i + 60Cos(theta)i + 60Sin(theta)j = 0
Cos(theta) = 5/60
theta = 84.29

-5i + 60Cos(84.29)i = -54.07i
Vb = -54.07i + 0j
t = 0.112hrs
t = 402.068s

To complete one circuit, the boat should be steered at an angle of 6.77 degrees from A to B, -9.46 degrees from B to C, and 84.29 degrees from C to A. The total time to complete one circuit would be 956.204 seconds, or approximately 15.94 minutes.

As a scientist, it is important to consider the accuracy and precision of the measurements and calculations used in solving this problem. The given values for the boat's cruising speed and the current's velocity should be carefully measured and accounted for in the calculations. Additionally, factors such as wind and other external forces may impact the boat's actual speed and direction, so these should also be taken into consideration. It may also be beneficial to conduct multiple trials and average the results to account for any potential errors.