How Do Wind and Plane Speeds Combine to Affect Ground Velocity?

[rossel]

Hi, I am having troubles with a question concerning relative motion. The problem goes:

A plane is traveling at an air speed of 285 km/h [E45ºS]. A wind is blowing to the northeast at 75 km/h [N22ºE] relative to the ground. Determine the velocity relative to the ground.

My teach prefers that we do not use the cosine law to determine the unknown vector, and instead requests that when we draw our diagram, we do something called the resolution of vectors. Which is separating the two vectors into x and y components, add the x's and y's together and use Pythagorean theorum to solve.

The cosine law provided me with the correct answer for this question, which was 265 km/h.

However, in using his method, my diagram (which was correct, and I have attached, to anyone who is confused), resembled a rectangle, and I came up with these calculations:

285/sin90 = x1/sin45
x1 = 202 km/h

x2/sin22 = 75/sin90
x2 = 28 km/h

Therefore, shouldn't x1 + x2 = 265 km/h ?
Please help me!

 

[learningphysics]

Yeah, I'm confused by your diagram. Your directions don't look right (which way on the diagram is north?). Plus you haven't drawn the vectors as being added.

To add vectors... suppose your adding a vector B to vector A... The tail (part without the arrow) of B has to be placed at the head (part with the arrow) of A.

Draw the diagram again.

 

[m2003]

Hi there,

Vectors and relative motion can be tricky, but with practice and a solid understanding of the concepts, you will be able to solve problems like this easily.

First, let's define some terms. A vector is a quantity that has both magnitude (size) and direction. In this problem, the plane's air speed and the wind's speed are both vectors. Relative motion is the motion of an object with respect to another object or frame of reference. In this problem, we are trying to find the plane's velocity relative to the ground, which is the frame of reference.

Now, let's discuss the method your teacher prefers - the resolution of vectors. This method involves breaking down the given vectors into their x and y components, adding them together, and then using the Pythagorean theorem to find the resultant vector. This method is based on the concept that vectors can be added together using the head-to-tail method, where the head of one vector is connected to the tail of the other vector. This results in a parallelogram, which can be broken down into two right triangles. The x and y components are simply the sides of these triangles.

In your diagram, you correctly separated the vectors into their x and y components. However, when adding the x components, you should use the cosine law, not the sine law. This is because the x components are not perpendicular to each other, so we cannot use the sine law. The correct equation would be:

x1 + x2 = 285 cos 45 + 75 cos 22 = 265 km/h

Similarly, for the y components, we would use the sine law:

y1 + y2 = 285 sin 45 + 75 sin 22 = 160 km/h

Now, we can use the Pythagorean theorem to find the resultant vector:

v = √(x^2 + y^2) = √(265^2 + 160^2) = 315 km/h [E34ºS]

This is the same answer we got using the cosine law, but using the resolution of vectors method. So, both methods are valid and will give you the correct answer, as long as you use the correct equations and methods for each step.

I hope this helps clarify the concept of vectors and relative motion for you. Keep practicing and remember to always pay attention to the direction and magnitude of the given vectors. Good luck!