A wind is blowing at 30 degree west of north at 20km/hr. A bird is flying in the wind and its velocity relative to the ground is 90km/hr west of north. Calculate the velocity and direction of the bird
what direction is "west of North"?
Answer
To calculate the velocity and direction of the bird, we can break down the velocities into their north and west components.
Let's define the following directions:
- North: positive
- West: negative
Wind velocity relative to the ground:
- Wind angle = 30 degrees west of north
- Wind speed = 20 km/hr
Bird velocity relative to the ground:
- Bird angle = 90 degrees west of north
- Bird speed = 90 km/hr
First, we calculate the west component of the wind velocity:
West component = Wind speed * cos(Wind angle)
= 20 km/hr * cos(30 degrees)
≈ 20 km/hr * 0.866
≈ 17.32 km/hr (west)
Next, we calculate the north component of the wind velocity:
North component = Wind speed * sin(Wind angle)
= 20 km/hr * sin(30 degrees)
≈ 20 km/hr * 0.5
≈ 10 km/hr (north)
Now, we can find the west component of the bird velocity:
West component = Bird speed * cos(Bird angle)
= 90 km/hr * cos(90 degrees)
= 0 (since cos(90 degrees) = 0)
Finally, we find the north component of the bird velocity:
North component = Bird speed * sin(Bird angle)
= 90 km/hr * sin(90 degrees)
= 90 km/hr (north)
To calculate the total velocity and direction of the bird, we add the north and west components together:
Total velocity = sqrt((west component)^2 + (north component)^2)
= sqrt((0)^2 + (90 km/hr)^2)
= sqrt(0 + 8100 km^2/hr^2)
= sqrt(8100) km/hr
= 90 km/hr
Since the west component is 0, we know that the bird's direction is purely north.
So, the velocity of the bird is 90 km/hr and the direction is north.
To calculate the velocity and direction of the bird, we first need to break down the given information into its components.
Velocity of the wind:
- The wind is blowing at 30 degrees west of north.
- The magnitude of the wind's velocity is given as 20 km/hr.
From this information, we can calculate the components of the wind's velocity using trigonometry. We'll assume that the north direction is the positive y-axis and the west direction is the negative x-axis.
Components of the wind's velocity:
Wind velocity along the x-axis = 20 km/hr * sin(30°)
Wind velocity along the y-axis = 20 km/hr * cos(30°)
Next, let's consider the velocity of the bird relative to the ground:
- The bird's velocity is given as 90 km/hr west of north.
Similar to above, we can express the bird's velocity in terms of its components along the x and y-axis.
Components of the bird's velocity:
Bird velocity along the x-axis = 90 km/hr * sin(30°)
Bird velocity along the y-axis = 90 km/hr * cos(30°)
To determine the combined velocity of the bird and the wind, we add up the respective components along the x and y-axis:
Combined velocity along the x-axis = Wind velocity along the x-axis + Bird velocity along the x-axis
Combined velocity along the y-axis = Wind velocity along the y-axis + Bird velocity along the y-axis
The magnitude and direction of the bird's velocity relative to the ground can be calculated using the Pythagorean theorem and trigonometry:
Magnitude of combined velocity = sqrt( (Combined velocity along the x-axis)^2 + (Combined velocity along the y-axis)^2 )
Direction of combined velocity = arctan( (Combined velocity along the y-axis) / (Combined velocity along the x-axis) )
By calculating the above values, we can find the velocity and direction of the bird relative to the ground.