A ship that can cruise at 12km/h in still waters , sets course due S.W .it is driven off by current flowing W.15.N at a rate of 4 km/h.calulate : a. The resultant velocity
b. The displacement after 3 hours.
I will do this one using vectors.
I will also use the convention: due S.W ----> 225°, W.15.N ----> 165°
vector R = (12cos225°, 12sin225°) + (4cos165°, 4sin165°
= (-8.485..., -8.485...) + (-3.863..., 1.0353..) = (-12.348..., -7.45...)
|R| = √((-12.348..)^2 + (-7.45..)^2 ) = appr 14.4
So the velocity is 14.4 km/h
direction: tanØ = -7.45/-12.348 , Ø = (180+31.1)° = appr 211.1° , or using your notation W 31.1° S
I am sure you can find the answer to the last part of your problem
To calculate the resultant velocity, we can use vector addition. The ship's velocity (in still water) is 12 km/h due S.W, and the current's velocity is 4 km/h due W.15.N.
a. To find the resultant velocity, we need to add these two vectors together.
Step 1: Convert the current's direction to components.
The current velocity is due W.15.N, which means it has a westward component and a northward component.
The westward component can be calculated as follows:
Westward component = Current velocity * sin(angle)
Westward component = 4 km/h * sin(15°)
The northward component can be calculated as follows:
Northward component = Current velocity * cos(angle)
Northward component = 4 km/h * cos(15°)
Step 2: Add the ship's velocity vector and the current's velocity vector together.
The ship's velocity vector has a magnitude of 12 km/h.
The resultant velocity vector can be calculated as follows:
Resultant westward component = ship's westward component + current's westward component
Resultant northward component = ship's northward component + current's northward component
Resultant westward component = 12 km/h * cos(225°) + (4 km/h * sin(15°))
Resultant northward component = 12 km/h * sin(225°) + (4 km/h * cos(15°))
Step 3: Calculate the magnitude and direction of the resultant velocity vector.
Magnitude = sqrt(Resultant westward component^2 + Resultant northward component^2)
Direction = arctan(Resultant northward component / Resultant westward component)
b. To calculate the displacement after 3 hours, multiply the resultant velocity by the time.
Displacement = Resultant velocity * Time
Let's calculate these values step by step.
a. The resultant velocity:
Resultant westward component = 12 km/h * cos(225°) + (4 km/h * sin(15°))
Resultant westward component ≈ -0.586 km/h + 1.067 km/h
Resultant westward component ≈ 0.481 km/h
Resultant northward component = 12 km/h * sin(225°) + (4 km/h * cos(15°))
Resultant northward component ≈ -10.392 km/h + 3.454 km/h
Resultant northward component ≈ -6.938 km/h
Magnitude = sqrt(Resultant westward component^2 + Resultant northward component^2)
Magnitude ≈ sqrt((0.481 km/h)^2 + (-6.938 km/h)^2)
Magnitude ≈ sqrt(0.231 km^2/h^2 + 48.179 km^2/h^2)
Magnitude ≈ sqrt(48.410 km^2/h^2)
Magnitude ≈ 6.958 km/h
Direction = arctan(Resultant northward component / Resultant westward component)
Direction = arctan(-6.938 km/h / 0.481 km/h)
Direction ≈ arctan(-14.407)
Direction ≈ -84.25°
Therefore, the resultant velocity is approximately 6.958 km/h due S.W., at an angle of -84.25°.
b. The displacement after 3 hours:
Displacement = Resultant velocity * Time
Displacement = 6.958 km/h * 3 hours
Displacement ≈ 20.874 km
Therefore, the displacement after 3 hours is approximately 20.874 km.
To calculate the resultant velocity and displacement, we need to consider the vectors involved. Let's break down the information given:
- The ship can cruise at 12 km/h in still waters. This means that if there were no external forces or currents, the ship would travel at a speed of 12 km/h in any direction.
- The ship sets course due southwest (S.W.). This means that the ship is initially moving in the southwest direction.
- The ship is driven off by a current flowing west-northwest (W.15.N). This means that there is a current moving from west to northwest.
- The rate of the current is 4 km/h.
Now, let's calculate the resultant velocity and displacement:
a. Resultant Velocity:
To find the resultant velocity, we need to add the velocity of the ship to the velocity of the current. Since velocity is a vector quantity, we need to consider both speed and direction.
Since the ship is moving southwest, its velocity can be represented as 12 km/h at an angle of 225 degrees (measured counterclockwise from the positive x-axis). To add the velocity of the current, we need to convert the current direction (W.15.N) to a vector.
To convert the current direction, we can use vectors. "W.15.N" means that the current is coming from the west (270 degrees) and is diverted 15 degrees north of west (15 degrees counterclockwise from the west). We can represent it as a vector with a magnitude of 4 km/h and a direction of 255 degrees (270 degrees - 15 degrees).
To find the resultant velocity, we can use vector addition. We add the x-components and y-components of the two vectors separately and calculate the magnitude and direction of the resulting vector.
Using the vectors' magnitude multiplied by their respective cosines and sines, we get:
x-component of ship velocity = 12 km/h * cos(225 degrees) = -8.49 km/h
y-component of ship velocity = 12 km/h * sin(225 degrees) = -8.49 km/h
x-component of current velocity = 4 km/h * cos(255 degrees) = -3.76 km/h
y-component of current velocity = 4 km/h * sin(255 degrees) = -1.05 km/h
Adding the x-components and y-components separately:
x-component of resultant velocity = -8.49 km/h - 3.76 km/h = -12.25 km/h (negative means westward direction)
y-component of resultant velocity = -8.49 km/h - 1.05 km/h = -9.54 km/h (negative means southward direction)
To find the magnitude of the resultant velocity (speed), we use the Pythagorean theorem:
resultant velocity = sqrt((-12.25 km/h)^2 + (-9.54 km/h)^2) = 15.9 km/h (rounded to one decimal place)
The direction of the resultant velocity can be found using trigonometry:
angle = atan(-9.54 km/h / -12.25 km/h) = 37.06 degrees (rounded to two decimal places)
So, the resultant velocity is 15.9 km/h at an angle of 37.06 degrees (measured counterclockwise from the positive x-axis), in the southwest direction.
b. Displacement after 3 hours:
To calculate displacement, we multiply the resultant velocity by the time.
displacement = resultant velocity * time
Substituting the values, we get:
displacement = 15.9 km/h * 3 hours = 47.7 km (rounded to one decimal place)
So, the displacement after 3 hours is 47.7 km.