Two sailing boats are competing in a race. The first boat (the Cutty Sark) has a velocity given by v(t) = 3j. It is travelling due North.

The second boat (the Marie Celeste) has a velocity given by v(t) =31 + 4j.
You are lucky enough to be on the Cutty Sark. You observe the Marie Celeste and determine its
velocity relative to your boat. All units are SI units.
i. Calculate the speed of the Marie Celeste relative to the Cutty Sark.
ii. Calculate the direction of the Marie Celeste relative to the Cutty Sark.
(Give your answer as degrees from North, or draw a sketch to carefully indicate the
direction.)

To find the speed and direction of the Marie Celeste relative to the Cutty Sark, we first need to understand the velocities of both boats. The Cutty Sark is moving due North with a velocity vector of v(t) = 3j, while the Marie Celeste has a velocity vector of v(t) = 3i + 4j. Let's break this down step by step.

Step 1: Determine the Relative Velocity

The relative velocity of the Marie Celeste with respect to the Cutty Sark can be calculated by subtracting the velocity of the Cutty Sark from that of the Marie Celeste. Mathematically, this is expressed as:

v_relative = v_Marie_Celeste - v_Cutty_Sark

Substituting the given velocities:

v_relative = (3i + 4j) - (0i + 3j) = 3i + (4j - 3j) = 3i + 1j

Step 2: Calculate the Speed of the Marie Celeste Relative to the Cutty Sark

The speed of the relative velocity vector can be found using the Pythagorean theorem. The speed is the magnitude of the vector:

Speed = √(vx² + vy²)

Here, vx = 3 and vy = 1. Plugging in these values:

Speed = √(3² + 1²) = √(9 + 1) = √10

Calculating this gives:

Speed ≈ 3.16 m/s

Step 3: Determine the Direction of the Marie Celeste Relative to the Cutty Sark

To find the direction, we need to calculate the angle θ that the relative velocity vector makes with the North direction (the positive y-axis). We can use the tangent function:

tan(θ) = opposite/adjacent = vy/vx

In this case, vy = 1 and vx = 3:

tan(θ) = 1/3

Now, we can find θ by taking the arctangent:

θ = arctan(1/3)

Calculating this gives:

θ ≈ 18.43 degrees

Final Results

To summarize:

• Speed of the Marie Celeste relative to the Cutty Sark: Approximately 3.16 m/s
• Direction of the Marie Celeste relative to the Cutty Sark: Approximately 18.43 degrees East of North

So, while you're sailing on the Cutty Sark, you can see that the Marie Celeste is moving at a speed of about 3.16 m/s at an angle of about 18.43 degrees to the east of your northward path. This means it's slightly veering off to the east as it races along!