Consider an airplane headed N 60 degrees W at a speed of 400 mph. What wind direction at a speed of 75 mph will produce a resultant direction of N 50 degrees W?
To solve this problem, we need to use vector addition. The airplane's velocity and the wind velocity will combine to produce the resultant velocity, which is the direction and speed of the airplane relative to the ground.
Let's break down the problem step by step:
1. Start by drawing a diagram to represent the situation. Draw a line to represent the airplane's velocity, N 60 degrees W, and another line to represent the wind's velocity, which we will label as "W."
2. Label the magnitude (speed) and direction of the airplane's velocity: 400 mph at N 60 degrees W. The magnitude represents the length of the line, and the direction is given by the angle in relation to the reference (North).
3. Label the magnitude (speed) and direction of the resultant velocity (airplane + wind): Unknown speed at N 50 degrees W. We will label this line as "V" to represent the resultant velocity.
4. Note that the direction N 50 degrees W is counterclockwise from West (which is 270 degrees) by 50 degrees. So, N 50 degrees W can also be represented as 270 degrees - 50 degrees, which equals 220 degrees.
5. The resultant velocity vector (V) is the sum of the airplane velocity vector (A) and the wind velocity vector (W). Mathematically, we can represent it as V = A + W.
6. Convert the given velocities (A and W) into their components using trigonometry. For the airplane's velocity (A), the North-South component = 400 mph * sin(60 degrees) and the West-East component = 400 mph * cos(60 degrees). Similarly, for the wind's velocity (W), the North-South component = 75 mph * sin(x) and the West-East component = 75 mph * cos(x), where 'x' represents the wind direction (unknown).
7. Equate the components to find the resultant components. The North-South component of the resultant velocity (V) should equal the sum of the North-South components of A and W, and the West-East component of V should equal the sum of the West-East components of A and W.
8. Setting up the equations:
- North-South component: 400 mph * sin(60 degrees) + 75 mph * sin(x) = V_north
- West-East component: 400 mph * cos(60 degrees) + 75 mph * cos(x) = V_east
9. Now, substitute the given resultant direction, N 50 degrees W, which we converted to 220 degrees, into the equations.
- North-South component: 400 mph * sin(60 degrees) + 75 mph * sin(x) = V_north
- West-East component: 400 mph * cos(60 degrees) + 75 mph * cos(x) = V_east
10. Solve these two equations simultaneously to find the wind direction, which is represented by 'x.' Substitute the values into the equation and solve for 'x'.
Once you find the value of 'x,' you will have the wind direction that, combined with its speed of 75 mph, will produce a resultant direction of N 50 degrees W.