A ship is heading due north at 12km/h in a tide moving 5km/h westward. Determine the magnitude and direction of the resultant velocity of the ship?
speed = sqrt (144 + 25)
= 13
tan angle north of west = 12/5
so angle = 67.4
so
west 67.4 north
or north 22.6 west
or
337.4 on your compass
Well, I suppose the ship is really feeling the tug of the tide, like a strong-arm pulling it to the west. It's like the tide wants to be its new BFF, but the ship is not quite ready for that level of commitment.
Let's get down to business and calculate the magnitude and direction of the ship's resultant velocity. To find the magnitude, we need to first find the horizontal and vertical components of the velocity.
Given that the ship is heading due north at 12 km/h, we can say that its vertical component is 12 km/h. And since the tide is moving westward at 5 km/h, the horizontal component is 5 km/h.
To find the resultant velocity, we can use the Pythagorean theorem. The magnitude of the resultant velocity will be the square root of the sum of the squares of the horizontal and vertical components.
Magnitude = sqrt((12 km/h)^2 + (5 km/h)^2)
And the direction? It's going to be like a compass needle pointing in the direction of the resultant velocity. In other words, we'll use trigonometry to find the angle.
Direction = arctan(vertical component/horizontal component)
Now, grab your calculator and let the giggles begin!
To determine the magnitude and direction of the resultant velocity of the ship, we can use vector addition.
1. First, draw a diagram to represent the two velocities: the ship's velocity heading due north and the tide's velocity moving westward.
- The ship's velocity is represented by an arrow pointing straight up (north) with a length of 12 km/h.
- The tide's velocity is represented by an arrow pointing to the left (west) with a length of 5 km/h.
2. Next, draw a vector diagram by placing the tail of the second arrow (tide's velocity) at the head of the first arrow (ship's velocity).
3. Connect the tail of the first arrow to the head of the second arrow with a third arrow to represent the resultant velocity.
4. Measure the length of the third arrow, which represents the magnitude of the resultant velocity. In this case, the length of the third arrow can be measured to be approximately 13 km/h using a ruler or by visually comparing the lengths.
5. Finally, determine the direction of the resultant velocity by measuring the angle between the resultant velocity arrow and the northern direction. In this case, the angle can be measured to be approximately 22.6 degrees counterclockwise from the north direction using a protractor or by visually estimating the angle.
Therefore, the magnitude of the resultant velocity is approximately 13 km/h, and the direction of the resultant velocity is approximately 22.6 degrees counterclockwise from the north direction.
To solve this problem, we can use vector addition. The velocity of the ship consists of two components: its own velocity due north and the velocity of the tide moving westward.
Let's represent the velocity of the ship due north as Vn and the velocity of the tide moving westward as Vw.
The magnitude of Vn is given as 12 km/h and its direction is due north. So, Vn = 12 km/h in the direction of north.
The magnitude of Vw is given as 5 km/h, and its direction is westward. So, Vw = 5 km/h in the direction of west.
To find the resultant velocity, we need to add the vectors Vn and Vw using vector addition.
Here's the step-by-step process to find the resultant velocity:
Step 1: Convert the velocities to vectors:
Vn = 12 km/h (due north) = 12 km/h (0°)
Vw = 5 km/h (westward) = 5 km/h (270°)
Step 2: Use vector addition to find the resultant velocity:
Add the horizontal (x) components and the vertical (y) components of Vn and Vw separately.
The x-component of Vn is 12 km/h * cos(0°) = 12 km/h * 1 = 12 km/h
The y-component of Vn is 12 km/h * sin(0°) = 12 km/h * 0 = 0 km/h
The x-component of Vw is 5 km/h * cos(270°) = 5 km/h * 0 = 0 km/h
The y-component of Vw is 5 km/h * sin(270°) = 5 km/h * (-1) = -5 km/h
Step 3: Add the x-components and y-components:
Resultant x-component = 12 km/h + 0 km/h = 12 km/h
Resultant y-component = 0 km/h + (-5 km/h) = -5 km/h
Step 4: Calculate the magnitude and direction of the resultant velocity:
The magnitude (R) of the resultant velocity is given by the Pythagorean theorem:
R = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
= sqrt((12 km/h)^2 + (-5 km/h)^2)
= sqrt(144 km^2/h^2 + 25 km^2/h^2)
= sqrt(169 km^2/h^2)
= 13 km/h (approx)
The direction (θ) of the resultant velocity can be found using the inverse tangent function:
θ = atan(Resultant y-component / Resultant x-component)
= atan(-5 km/h / 12 km/h)
≈ -22.6°
Therefore, the magnitude of the resultant velocity of the ship is 13 km/h, and its direction is approximately 22.6° west of due north.