What is the speed of the rocket relative to the Earth?

[terryds]

Homework Statement

Plane A flies with speed 0.6c chasing plane B which speed is 0.4c . Both speed is measured by observer on Earth. Then, plane B fires a small rocket which rest mass is 10 kg towards plane A. Rocket speed is 0.2c relative to plane B where c equals the speed of light in vacuum.
What's the speed of the rocket relative to the Earth?

Homework Equations

$V = \frac{V' + u}{1 + \frac{V' u}{c^2}}$ (reverse Lorentz transformation)

The Attempt at a Solution

So, I think the stationary frame is the Earth. The moving frame is plane B. The event is the rocket.

I put
V' = -0.2 c (because A chases B, then B fires a rocket towards A which means opposite direction of the plane), u = 0.4 c (because the moving frame is plane B, I define positive direction is the direction of the plane)

But, I get V = 0.217 c which means that the rocket has the same direction to those planes according to the observer in the Earth.
I think it should be negative sign.
Please help me where I got wrong.

The solution is -0.56 c but I don't know how to figure it out

 

[Orodruin]

Please help me where I got wrong.

Easy. You are not wrong. Well, except for:

I think it should be negative sign.

From the Earth frame, the small rocket will move in the same direction as B, since the relative speed between B and the rocket is smaller than the relative speed between B and the Earth frame.

 

[terryds]

Easy. You are not wrong. Well, except for:

From the Earth frame, the small rocket will move in the same direction as B, since the relative speed between B and the rocket is smaller than the relative speed between B and the Earth frame.

So, the solution is wrong, right? I also doubt the book since it's just written by my seniors hahaha.. thank you very much