A ship is being pushed by the sea currents at 25 km/hr 28 degrees west of south. find the velocity of the ship to the west and to the south
Let the current velocity be V km/hr
and θ be the angle west of south, then, using trigonometry,
The west component is Vsin(θ)
and the south component is Vcos(θ)
To find the velocity of the ship to the west and to the south, we need to break down the given current velocity into its westward and southward components.
First, we'll find the westward (x) component:
Velocity in the west direction = Current velocity * cos(angle)
Westward component = 25 km/hr * cos(28°)
Westward component = 22.61 km/hr
Next, we'll find the southward (y) component:
Velocity in the south direction = Current velocity * sin(angle)
Southward component = 25 km/hr * sin(28°)
Southward component = 11.86 km/hr
Therefore, the velocity of the ship to the west is 22.61 km/hr, and the velocity of the ship to the south is 11.86 km/hr.
To find the velocity of the ship to the west and to the south, we can break down the given velocity of the ship into its westward (x-) and southward (y-) components.
Given:
Speed = 25 km/hr
Direction = 28 degrees west of south
Step 1: Convert the direction to a standard mathematical angle.
Since the given angle is west of south, we need to convert it to an angle measured from the positive x-axis (east). To do this, subtract the given angle from 90 degrees:
90 degrees - 28 degrees = 62 degrees
Now we have an angle of 62 degrees measured from the positive x-axis.
Step 2: Calculate the westward (x-) and southward (y-) components.
To find the x- and y-components, we can use the trigonometric functions cosine (cos) and sine (sin).
The x-component (westward) can be calculated using:
x-component = speed * cos(angle)
The y-component (southward) can be calculated using:
y-component = speed * sin(angle)
Calculating the components:
x-component = 25 km/hr * cos(62 degrees)
y-component = 25 km/hr * sin(62 degrees)
Step 3: Calculate the values of the x- and y-components.
Using a scientific calculator, calculate the values of the cosine and sine functions for the given angle:
cos(62 degrees) ≈ 0.442
sin(62 degrees) ≈ 0.897
Now, substitute these values into the formulas to calculate the x- and y-components:
x-component = 25 km/hr * 0.442
y-component = 25 km/hr * 0.897
Step 4: Simplify the calculations.
Multiply the speed value by the corresponding trigonometric function values:
x-component ≈ 11.05 km/hr (rounded to two decimal places)
y-component ≈ 22.43 km/hr (rounded to two decimal places)
Step 5: Interpret the results.
The velocity of the ship to the west is approximately 11.05 km/hr (rounded to two decimal places), and the velocity of the ship to the south is approximately 22.43 km/hr (rounded to two decimal places).