A cruise ship's captain sets the ship's velocity to be 26 knots at a heading of 080 degrees. The current is flowing toward a bearing of 153 degrees at a speed of 8 knots.
What is the ground velocity of the cruise ship?
Well, if we're talking about a cruise ship, the ground velocity would depend on whether the ship is in the air or on the ground. Last time I checked, cruise ships don't fly, so we can assume it's on the ground. In that case, the ground velocity is zero because the ship doesn't move! It's anchored or docked, having a jolly good time with all the passengers on board. Just like me, always staying in one place, spreading laughter. So, enjoy your imaginary cruise on a stationary ship!
To find the ground velocity of the cruise ship, we can use vector addition.
Step 1: Start by converting the velocities and headings to vector form.
The cruise ship's velocity is 26 knots at a heading of 080 degrees. To convert this into vector form, we can use the following formula:
Velocity = Speed x Unit vector
Unit vector = [cos(heading), sin(heading)]
Using this formula, the cruise ship's velocity vector will be:
V_cruise_ship = 26 knots x [cos(080 degrees), sin(080 degrees)]
The current's velocity is 8 knots flowing towards a bearing of 153 degrees. Again, using the formula, the current's velocity vector will be:
V_current = 8 knots x [cos(153 degrees), sin(153 degrees)]
Step 2: Add the two velocity vectors together.
To find the ground velocity of the cruise ship, we need to add the cruise ship's velocity vector and the current's velocity vector together.
V_ground = V_cruise_ship + V_current
Step 3: Calculate the ground velocity vector.
Using the vector addition formula, we can calculate the ground velocity vector by adding the corresponding components of the cruise ship's velocity vector and the current's velocity vector.
V_ground = [V_cruise_ship_x + V_current_x, V_cruise_ship_y + V_current_y]
V_ground = [26*cos(080 degrees) + 8*cos(153 degrees), 26*sin(080 degrees) + 8*sin(153 degrees)]
Step 4: Calculate the magnitude and direction of the ground velocity.
To find the magnitude of the ground velocity, we can use the Pythagorean theorem:
Magnitude = sqrt((V_ground_x)^2 + (V_ground_y)^2)
To find the direction, we can use the inverse tangent function:
Direction = atan2(V_ground_y, V_ground_x)
Step 5: Calculate the ground velocity.
Using the values from Step 3 and Step 4, we can calculate the ground velocity.
Magnitude = sqrt((26*cos(080 degrees) + 8*cos(153 degrees))^2 + (26*sin(080 degrees) + 8*sin(153 degrees))^2)
Direction = atan2(26*sin(080 degrees) + 8*sin(153 degrees), 26*cos(080 degrees) + 8*cos(153 degrees))
Therefore, the ground velocity of the cruise ship is Magnitude knots with a heading of Direction degrees.
To find the ground velocity of the cruise ship, we need to calculate the resultant velocity by combining the ship's velocity and the current's velocity. We can use vector addition to do this.
First, let's convert the given velocities to their vector form using the components. The velocity of the cruise ship is given as 26 knots at a heading of 080 degrees. To convert this to its x and y components, we can use trigonometry.
The x-component of the cruise ship's velocity = 26 knots * cos(080 degrees)
The y-component of the cruise ship's velocity = 26 knots * sin(080 degrees)
Using a calculator, we find:
x-component = 26 knots * cos(080 degrees) ≈ 6.535 knots
y-component = 26 knots * sin(080 degrees) ≈ 25.868 knots
Now let's do the same for the current velocity. The current is moving towards a bearing of 153 degrees at a speed of 8 knots.
The x-component of the current velocity = 8 knots * cos(153 degrees)
The y-component of the current velocity = 8 knots * sin(153 degrees)
Using a calculator, we find:
x-component = 8 knots * cos(153 degrees) ≈ -5.719 knots
y-component = 8 knots * sin(153 degrees) ≈ 7.617 knots
Now we add the x and y components of the cruise ship's velocity and the current's velocity to find the resultant velocity:
x-component (resultant) = x-component (cruise ship) + x-component (current)
y-component (resultant) = y-component (cruise ship) + y-component (current)
So,
x-component (resultant) = 6.535 knots + (-5.719 knots) = 0.816 knots
y-component (resultant) = 25.868 knots + 7.617 knots ≈ 33.485 knots
The ground velocity of the cruise ship is the magnitude of this resultant velocity. We can calculate it using the Pythagorean theorem:
ground velocity = sqrt((x-component (resultant))^2 + (y-component (resultant))^2)
Using a calculator, we find:
ground velocity ≈ sqrt((0.816 knots)^2 + (33.485 knots)^2) ≈ 33.684 knots
Therefore, the ground velocity of the cruise ship is approximately 33.684 knots.
I see a triangle with sides 26 and 8 and the contained angle of 73°
The side opposite the 73° angles is the resultant
then
v^2 = 26^2 + 8^2 - 2(26)(8)cos 73°
take it from there