This is my homework problem:
"The pilot of a plane is flying at 450 mi/hr W 20.0 degrees S relative to the earth. She sees another plane flying at 350 mi/hr N 40.0 degrees W relative to her plane. What was the magnitude and direction of the plane she saw relative to the earth?"
I'm not entirely sure where to start because relative motion is something I'm still trying to grasp. What do I start with to find the magnitude?
Velocity of the other plane with respect to Earth =
Velocity of the other plane w.r.t. her plane
+ Velocity of her plane w.r.t. Earth
Perform the vector addition
That doesn't require me to find the x and y components of those vectors, does it?
Yes it does. It's not hard.
Call North the +y direction and east the +x direction.
It is not clear to me if "W 20.0 degrees S" means "20 degrees west of south" or "20 degrees south of west". You will need to clarify the terminology to answer the question.
So, would the components look something like this:
450cos200
450sin200
350cos130
350sin130
To find the magnitude and direction of the plane the pilot saw relative to the earth, we can break down the problem into two components: finding the horizontal component and the vertical component of the velocity.
1. Start by creating a coordinate system: Let's assume that the positive x-direction is East and the positive y-direction is North, as is common convention.
2. The first step is to convert the given speeds and angles into their respective components. Use trigonometry to separate the velocities into horizontal and vertical components.
For the pilot's plane:
- Velocity: 450 mi/hr at 20.0 degrees south of west
- Horizontal component: 450 mi/hr * cos(20.0 degrees)
- Vertical component: 450 mi/hr * sin(20.0 degrees)
For the other plane seen:
- Velocity: 350 mi/hr at 40.0 degrees west of north
- Horizontal component: 350 mi/hr * sin(40.0 degrees)
- Vertical component: 350 mi/hr * cos(40.0 degrees)
3. Next, add the horizontal components and the vertical components separately to find the resultant horizontal and vertical velocities.
- Resultant horizontal velocity: (Horizontal component of pilot's plane) + (Horizontal component of other plane seen)
- Resultant vertical velocity: (Vertical component of pilot's plane) + (Vertical component of other plane seen)
4. Finally, use the Pythagorean theorem to find the magnitude of the resultant velocity:
- Magnitude of velocity = sqrt((Resultant horizontal velocity)^2 + (Resultant vertical velocity)^2)
5. To find the direction of the resultant velocity, use trigonometry again:
- Direction = arctan(Resultant horizontal velocity / Resultant vertical velocity)
By following these steps, you should be able to find the magnitude and direction of the plane the pilot saw relative to the earth.