a ship is heading due north at 7.50 m/s relative to the water. The local current is 1.52 m/s in a direction 42 degree west of north. Find its velocity relative to earth( find both direction and magnitude of the velocity)
V=7.5 m/s
v=1.52 m/s
u=?
u(x)=V(x)+v(x)= 0-vsin42⁰= - 1.52•0.67=-1.02 m/s
u(y)=V(y)+v(y)= V+vcos42⁰=7.5+1.52•0.0.74=8.63 m/s
u=sqrt{u(x)²+ u(y)²}=8.69 m/s
tanβ = u(x)/u(y)=1.02/8.63=0.12
β =6.74⁰ (west of north)
To find the velocity of the ship relative to the Earth, we need to consider the vector addition of the ship's velocity relative to the water and the velocity of the current.
Step 1: Convert the given velocities to vector form.
The ship's velocity relative to the water is given as 7.50 m/s due north. This can be written as a vector, V_ship-water = 7.50 m/s north.
The current's velocity is given as 1.52 m/s in a direction 42 degrees west of north. We can find the vertical and horizontal components of the current's velocity using trigonometry. The vertical component is given by: V_current_vertical = 1.52 m/s * sin(42°), and the horizontal component is given by: V_current_horizontal = 1.52 m/s * cos(42°).
Step 2: Add the vectors.
To add vectors, we need to break them down into their horizontal (x-direction) and vertical (y-direction) components and then add them separately.
The x-component of the ship's velocity relative to the Earth is the sum of the x-components of the ship's velocity relative to the water and the current's horizontal velocity. This can be calculated as:
V_ship-earth_x = V_ship-water_x + V_current_horizontal
= 0 + 1.52 m/s * cos(42°)
The y-component of the ship's velocity relative to the Earth is the sum of the y-components of the ship's velocity relative to the water and the current's vertical velocity. This can be calculated as:
V_ship-earth_y = V_ship-water_y + V_current_vertical
= 7.50 m/s + 1.52 m/s * sin(42°)
Step 3: Find the magnitude and direction of the ship's velocity relative to Earth.
The magnitude of the ship's velocity relative to the Earth can be found using the Pythagorean theorem:
Magnitude = √(V_ship-earth_x² + V_ship-earth_y²)
The direction of the ship's velocity relative to the Earth can be calculated using trigonometry:
Direction = tan^(-1)(V_ship-earth_y / V_ship-earth_x)
By plugging in the values calculated above, you can find the magnitude and direction of the ship's velocity relative to the Earth.