Question text

At the beginning of a race, a yacht crosses the start line and sails in a westerly direction at a speed of 6 m s-1. The current of the sea is flowing at 2 m s-1 towards the north-east.
Calculate how farand in what direction (relative to west)the yacht has travelled after 18 minutes from its crossing the start line.

To solve this problem, we need to break it down into two components: the yacht's velocity and the current's velocity. The yacht's velocity is 6 m/s towards the west, and the current's velocity is 2 m/s towards the north-east.

Let's start by finding the yacht's distance traveled after 18 minutes. Since the yacht is traveling at a constant speed of 6 m/s, we can calculate the distance using the formula:

Distance = speed x time

Converting 18 minutes to seconds, we have:

Time = 18 minutes x 60 seconds/minute = 1080 seconds

So, the distance traveled by the yacht is:

Distance = 6 m/s x 1080 s = 6480 meters

Now let's calculate the north and east components of the current's velocity. The north component is given by the formula:

North Component = speed x sin(angle)

The angle between the current's direction (north-east) and the north direction is 45 degrees (since the angle between the north-east and east is 45 degrees). Let's calculate the north component of the current's velocity:

North Component = 2 m/s x sin(45 degrees) = 2 m/s x √2/2 = √2 m/s

Similarly, the east component is given by the formula:

East Component = speed x cos(angle)

Let's calculate the east component of the current's velocity:

East Component = 2 m/s x cos(45 degrees) = 2 m/s x √2/2 = √2 m/s

Now, we can find the total north and west components of the yacht's displacement. The west component is the distance traveled by the yacht, which is 6480 meters. Since the north component of the current's velocity is towards the yacht's north-west direction, we need to subtract it. The east component is towards the east direction, which is perpendicular to the west direction. Therefore, it has no effect on the yacht's final displacement to the west.

West Component = Distance - North Component = 6480 m - √2 m/s

Finally, we can calculate the magnitude and direction of the yacht's displacement. The magnitude is given by the Pythagorean theorem:

Magnitude = √(West Component^2 + East Component^2)

Direction = tan^(-1)(North Component / West Component)

Plugging in the values we calculated:

Magnitude = √((6480 m - √2 m/s)^2 + (√2 m/s)^2)

Direction = tan^(-1)(√2 m/s / (6480 m - √2 m/s))

By evaluating these formulas, you can find the final answer for how far the yacht has traveled and in what direction (relative to west) after 18 minutes from its crossing the start line.