At the beginning of a race, a yacht crosses the start line and sails in
westerly direction at a speed of 4 m s-1. The current of the sea is flowing at
2 m s-1 towards the north-east. Calculate how far [8 marks] and in what
direction (relative to west) [8 marks] the yacht has travelled after 10
minutes from its crossing the start line.
(-4,0) + (1.4,1.4) = (-2.6,1.4)
= 2.96 m/s at W28.3°N
After 10 minutes (600 seconds) the distance traveled is 1772m
To calculate how far and in what direction the yacht has traveled after 10 minutes, we can break down the yacht's velocity into two components: one in the west direction and one in the north direction.
First, let's convert the time from minutes to seconds. Since 1 minute is equal to 60 seconds, 10 minutes would be 10 * 60 = 600 seconds.
Next, we can calculate the distance traveled in the west direction. The yacht's velocity in the west direction is 4 m/s, and it has traveled for 600 seconds. Therefore, the distance traveled in the west direction is:
Distance_west = Velocity_west * Time = 4 m/s * 600 s = 2400 meters.
Now, let's calculate the distance traveled in the north direction. The yacht is also affected by the current, which is flowing at 2 m/s towards the north-east. We can resolve this current into its north and east components.
The north component of the current's velocity is given by:
Velocity_north = Velocity_current * sin(45°) = 2 m/s * sin(45°) ≈ 1.414 m/s.
To find the distance traveled in the north direction, we multiply the north component of the current's velocity by the time:
Distance_north = Velocity_north * Time = 1.414 m/s * 600 s ≈ 848.4 meters.
Now, we can find the total distance traveled by combining the distances in the west and north directions using the Pythagorean theorem:
Total Distance = √(Distance_west^2 + Distance_north^2) = √(2400^2 + 848.4^2) ≈ 2564.7 meters.
Finally, we can find the direction relative to the west by using trigonometry. The angle θ can be calculated as the arctan(Distance_north / Distance_west):
θ = arctan(Distance_north / Distance_west) = arctan(848.4 / 2400) ≈ 19.74°.
Therefore, the yacht has traveled approximately 2564.7 meters in a direction approximately 19.74° relative to the west.