A boat is traveling on a bearing of 25 degrees East of North at a speed of 4 knots (a knot is 1.852 km/h). After traveling for 3 hours, the boats heading is changed to due South and it travels for an additional 2 hours at 5 knots. Using a Vector diagram, what is the resultant?
See Sat. post.
To find the resultant of the boat's journey, we can break down the boat's motion into two components: the Northward component and the Eastward component.
First, let's calculate the distances traveled in each component.
Northward component:
Distance = speed x time = 4 knots x 3 hours = 12 knots
Eastward component:
Distance = speed x time = 4 knots x 3 hours x sin(25 degrees) = 4 knots x 3 hours x 0.423 = 5.076 knots
Since we have a positive Northward distance and a positive Eastward distance, the Northward component is positive and the Eastward component is positive.
Next, let's calculate the distances traveled in each component after the change in heading.
Southward component:
Distance = speed x time = 5 knots x 2 hours = 10 knots
Eastward component:
Distance = speed x time = 5 knots x 2 hours x cos(180 degrees - 25 degrees) = 5 knots x 2 hours x cos(155 degrees) = -7.641 knots
Since we have a negative Eastward distance, the Eastward component is negative.
Now, we can calculate the resultant distance in each component.
Northward component:
Resultant Northward distance = Northward distance after the heading change - Northward distance before the heading change
= (10 knots - 12 knots) = -2 knots
Eastward component:
Resultant Eastward distance = Eastward distance after the heading change + Eastward distance before the heading change
= (-7.641 knots + 5.076 knots) = -2.565 knots
Since we have a negative Northward distance and a negative Eastward distance, the resultant Northward component is negative and the resultant Eastward component is negative.
Using the Pythagorean theorem, we can calculate the magnitude of the resultant vector.
Resultant magnitude = sqrt((Resultant Northward distance)^2 + (Resultant Eastward distance)^2)
= sqrt((-2 knots)^2 + (-2.565 knots)^2)
= sqrt(4 knots^2 + 6.581 knots^2)
= sqrt(16 + 43.156)
= sqrt(59.156)
= 7.686 knots.
Therefore, the magnitude of the resultant vector is approximately 7.686 knots.
To calculate the resultant vector, we need to break down the boat's motion into its north-south (vertical) and east-west (horizontal) components. Then we can find the sum of these components to determine the overall direction and magnitude of the resultant vector.
First, let's calculate the initial north-south and east-west components of the boat's motion.
The boat is traveling on a bearing of 25 degrees East of North at a speed of 4 knots. To find the vertical component, we can use the sine function.
Vertical Component = Speed * sin(Bearing)
= 4 knots * sin(25 degrees)
≈ 1.69 knots
To find the horizontal component, we can use the cosine function.
Horizontal Component = Speed * cos(Bearing)
= 4 knots * cos(25 degrees)
≈ 3.62 knots
Now we have the initial components of the boat's motion:
Vertical Component = 1.69 knots (North direction)
Horizontal Component = 3.62 knots (East direction)
After 3 hours, the boat's heading is changed to due South. This means its bearing is now 180 degrees. The boat then travels at a speed of 5 knots.
The vertical component of the boat's motion is now:
Vertical Component = Speed * sin(Bearing)
= 5 knots * sin(180 degrees)
= 0 knots
The horizontal component of the boat's motion is now:
Horizontal Component = Speed * cos(Bearing)
= 5 knots * cos(180 degrees)
= -5 knots
The negative sign indicates that the component is in the opposite direction (West direction) to our reference point.
Now we have the final components of the boat's motion:
Vertical Component = 0 knots (South direction)
Horizontal Component = -5 knots (West direction)
To find the resultant vector, we can simply add the corresponding vertical and horizontal components together.
Resultant Vertical Component = Initial Vertical Component + Final Vertical Component
= 1.69 knots + 0 knots
= 1.69 knots (North direction)
Resultant Horizontal Component = Initial Horizontal Component + Final Horizontal Component
= 3.62 knots + (-5 knots)
= -1.38 knots (West direction)
Therefore, the resultant vector can be represented as:
Resultant = 1.69 knots (North direction) - 1.38 knots (West direction)
Using a vector diagram, we can draw an arrow from the starting point (0, 0) to the point (1.69, -1.38) to represent the resultant vector. The length of the arrow represents the magnitude of the resultant vector, and the direction of the arrow represents its direction.