A ship is heading North at 15 km/h in a tide moving a 6 km/h westward. Determine the magnitude and direction of the resultant velocity of the ship.
To determine the magnitude and direction of the resultant velocity of the ship, we can use vector addition.
Step 1: Draw a diagram
Start by drawing a diagram representing the situation. Draw a horizontal line to represent the westward direction and a vertical line to represent the northward direction. Label the ship's velocity as 15 km/h pointing north and the tide's velocity as 6 km/h pointing west.
Step 2: Break down the velocities into components
Since the velocities are given in different directions, we need to break them down into their horizontal (x) and vertical (y) components.
For the ship, the vertical component (y-coordinate) is 15 km/h (moving north) and the horizontal component (x-coordinate) is 0 km/h (no movement westward).
For the tide, the vertical component (y-coordinate) is 0 km/h (no vertical movement) and the horizontal component (x-coordinate) is -6 km/h (moving westward).
Step 3: Add the x and y components separately
To find the resultant x-component, add the individual x-components of the ship and the tide:
0 km/h + (-6 km/h) = -6 km/h
To find the resultant y-component, add the individual y-components of the ship and the tide:
15 km/h + 0 km/h = 15 km/h
Step 4: Find the magnitude and direction of the resultant velocity
The magnitude of the resultant velocity is found using the Pythagorean theorem. The magnitude (V) is given by:
V = √(x^2 + y^2)
In this case, x = -6 km/h and y = 15 km/h:
V = √((-6 km/h)^2 + (15 km/h)^2)
V = √(36 km/h^2 + 225 km/h^2)
V = √261 km^2/h^2
Calculating V, the magnitude of the resultant is approximately 16.125 km/h (rounded to three decimal places).
The direction of the resultant velocity can be found using the inverse tangent function. The direction (θ) is given by:
θ = arctan(y/x)
In this case, y = 15 km/h and x = -6 km/h:
θ = arctan(15 km/h / -6 km/h)
θ ≈ -68.19°
The negative sign is because the direction is to the west (negative x-direction) from the north (positive y-direction). The direction can also be expressed as 291.81° (360° - 68.19°) from the positive x-axis in the counterclockwise direction.
So, the magnitude of the resultant velocity is approximately 16.125 km/h, and it is directed at an angle of approximately 291.81° (or -68.19°) from the positive x-axis.