To a motor cyclist traveling due North at 50km/h the wind appears to come from Northwest at 60km/h. What is the true velocity of the wind
To determine the true velocity of the wind, we need to find the vector sum of the motorcyclist's velocity and the wind velocity.
Let's break down the given information into vectors:
Motorcyclist's velocity (North) = 50 km/h
Wind velocity (Northwest) = 60 km/h
Since the wind is blowing from the Northwest, we can resolve it into its North and West components using trigonometry.
Given that the wind's velocity has a magnitude of 60 km/h, its North component can be found using the following equation:
North component = wind velocity * cos(angle)
Since the wind is blowing in a Northwest direction, the angle it makes with the North direction is 45 degrees. Therefore:
North component = 60 km/h * cos(45°)
North component = 60 km/h * √(2)/2
North component = 60√(2)/2 km/h
North component = 30√(2) km/h
To find the total velocity of the wind, we need to add the North and West components of the wind. Since the West component is 0 (the wind is not blowing in the East direction), the total velocity of the wind is equal to the North component alone.
Thus, the true velocity of the wind is 30√(2) km/h in the North direction.
To find the true velocity of the wind, we need to use vector addition.
Let's first represent the motorcyclist's velocity as a vector pointing due North at a magnitude of 50 km/h. We'll call this vector A.
Next, let's represent the apparent velocity of the wind as a vector pointing from the Northwest at a magnitude of 60 km/h. We'll call this vector B.
To find the true velocity of the wind, we can subtract vector A from vector B, which represents the wind's velocity relative to the ground.
Since A is pointing due North and B is pointing from the Northwest, we need to resolve B into its North and East components.
The Northwest direction is equivalent to a 45-degree angle relative to North, so we can use trigonometry to find the North and East components of vector B.
The North component of B can be calculated as follows:
North component of B = magnitude of B * cos(angle between B and North)
North component of B = 60 km/h * cos(45 degrees)
North component of B = 60 km/h * 0.7071
North component of B = 42.42 km/h (rounded to two decimal places)
The East component of B can be calculated as follows:
East component of B = magnitude of B * sin(angle between B and North)
East component of B = 60 km/h * sin(45 degrees)
East component of B = 60 km/h * 0.7071
East component of B = 42.42 km/h (rounded to two decimal places)
Now that we have the North and East components of B, we can subtract vector A (50 km/h due North) from the North and East components of B to find the true velocity of the wind:
North component of true wind velocity = North component of B - magnitude of A
North component of true wind velocity = 42.42 km/h - 50 km/h
North component of true wind velocity = -7.58 km/h
East component of true wind velocity = East component of B
East component of true wind velocity = 42.42 km/h
Therefore, the true velocity of the wind is approximately 7.58 km/h in the opposite direction of the motorcyclist's travel (South) and 42.42 km/h towards the East.