Relative Velocity Airplane Problem - Air and Ground speed

[Rock32]

Homework Statement

A jet flies with an airspeed of 310 mph due east (assuming +x is due east). It is climbing vertically with a velocity of 50 mph. A wind blowing toward northeast has a speed of 55 mph.

Homework Equations

Find the x,y,z components of the plane's velocity relative to the ground.

The Attempt at a Solution

Since nothing is affecting the z direction except the plane climbing, the z component would just be 50 mph.

Because the wind is blowing toward the northeast, this adds a y component of
(55 mph)*sin(45) =38.9 mph

Both those answers are correct, but I am stuck on the x component.

I tried adding 55mph*cos(45) to the 310 mph airspeed (since the plane is flying due east) to get 350 mph, but that is wrong.

I also tried using the airspeed as the final vector's magnitude (i.e. x^2 + y^2 + z^2 = 310) using the correct answers above for y and z, but that is also wrong.

Any help would be great.

 

[rl.bhat]

What is the exact value of x component of the velocity?

 

[kerimek]

As a scientist, it is important to approach this problem by breaking it down into its individual components and using vector addition to find the overall velocity of the plane relative to the ground.

First, we can visualize the situation by drawing a diagram with the x-axis representing east, the y-axis representing north, and the z-axis representing the vertical direction. The plane's velocity would be represented by a vector with a magnitude of 310 mph in the positive x direction.

Next, we can consider the wind as a separate vector with a magnitude of 55 mph and a direction of 45 degrees northeast. Using trigonometry, we can find the x and y components of this vector: 55 mph * cos(45) = 38.9 mph in the positive x direction and 55 mph * sin(45) = 38.9 mph in the positive y direction.

Now, we can use vector addition to find the overall velocity of the plane relative to the ground. This can be done by adding the individual components of the plane's velocity and the wind's velocity in each direction. The x component would be 310 mph + 38.9 mph = 348.9 mph, the y component would be 38.9 mph + 38.9 mph = 77.8 mph, and the z component would remain 50 mph.

Therefore, the overall velocity of the plane relative to the ground would be (348.9 mph, 77.8 mph, 50 mph). This means that the plane is moving at a speed of 348.9 mph in the east direction, 77.8 mph in the north direction, and 50 mph in the vertical direction.

In summary, it is important to break down the problem into its individual components and use vector addition to find the overall velocity of the plane relative to the ground. This approach can also be applied to more complex situations involving multiple velocities and directions.