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?
V1 = 4 * 1.852km/h = 7.408 km/h.[65o].
d1 = 3h * 7.408km/h = 22.224 km/h[65o].
V2 = 5 * 1.852km/h = 9.26 km/h.[270o].
d2 = 9.26km/h * 2h = 18.52 km[270o].
R = d1 + d2.
R = 22.224km[65o] + 18.52 km[270o]
X = 22.224*cos65 = 9.39 km.
Y = 22,224*sin65 + 18.52*sin270=1.62 km
tanA = Y/X = 1.62/9.39 = 0.17271
A = 9.8o
R = X/cosA = 9.39/cos9.8=9.53 km[9.8o].
= Resultant.
To calculate the resultant vector, we need to break down the boat's motion into its horizontal and vertical components and then add up those components separately.
1. For the initial motion, we have a bearing of 25 degrees East of North and a speed of 4 knots.
To convert this to horizontal and vertical components, we use trigonometry:
Horizontal component = speed * cos(bearing)
Vertical component = speed * sin(bearing)
So, the horizontal component = 4 knots * cos(25 degrees)
And the vertical component = 4 knots * sin(25 degrees)
2. After the change in heading, the boat travels due South at a speed of 5 knots for 2 hours.
The horizontal component in this case is zero because we're not moving horizontally.
The vertical component is simply the speed multiplied by the time:
Vertical component = 5 knots * 2 hours
3. Now we can add up the horizontal and vertical components to find the resultant vector.
Horizontal component of resultant = horizontal component from the initial motion + horizontal component from the second motion
Vertical component of resultant = vertical component from the initial motion + vertical component from the second motion
4. Finally, we can calculate the magnitude and direction of the resultant vector using Pythagoras' theorem and inverse trigonometric functions:
Magnitude = sqrt((horizontal component of resultant)^2 + (vertical component of resultant)^2)
Direction = arctan(vertical component of resultant / horizontal component of resultant)
By following these steps, you should be able to create a vector diagram and determine the resultant vector.