Kinetic Energy & Speed in Inertial Frames: Chris, Bob & Angelica

[Wout Veltman]

From Chris' perspective Bob is traveling with 1.5*10⁸ m/s in direction a. Angelica is also traveling with 2.4*10⁸ m/s in direction a.

From Bob's perspective Chris is traveling with 1.5*10⁸ m/s in direction b (The opposite of x). Angelica is traveling with 1.5*10⁸ m/s in direction a.

They all have a mass of 1

I am pretty sure these numbers are right. I used w = (u+v)/(1+(u*v)/(c²)) To calculate the relative speeds.
I used the calculation in the picture to calculate the E_k of Bob. I also calculated the E_k of Angelica, all from chris' perspective. Now the outcome that I was expecting was that Bob's E_k would be half of Angelica's E_k when looking from Chris' perspective. Because Angelica is also traveling with 1.5*10⁸ m/s when measured from Bob's perspective. Why does this not count up?

I am sorry if a skipped a few vital steps. All of my special relativity knowledge comes from self-studying. We don't get this in school.

Thank you in advance.

 

[beamie564]

Now the outcome that I was expecting was that Bob's Ek would be half of Angelica's Ek when looking from Chris' perspective

Why would that be true? Even in the non relativistic domain, it makes no sense.
Suppose you have an observer $C$ at the origin. An object $B$ moves with velocity $v$ in a particular direction. In the same direction, another object $A$ moves with velocity $v$ as seen by $B$. The velocity of $A$ as seen by $C$ will be $2v$ ( since this is non relativistic, they simply add). The kinetic energy ( as observed by $C$), of $A$ is not twice that of $B$, rather it's four times .
Likewise, in the relativistic domain, there is no reason for the results you expected.

 

[Wout Veltman]

I see I made a few unnecessary mistakes there, also in the picture I posted with it. But my confusion is still there, let me try to explain it in another way.

You have an observer $C$ at the origin. An object $B$ is moving with 1.5*10⁸ m/s in a particular direction. Object $B$ has his buddy $A$ moving next to him, with the same velocity and direction as $B$. Now object $A$ accelerates till he reaches a speed of 1.5*10⁸ m/s, when looking from $B$'s point of view, and a speed of 2.4*10⁸ m/s viewed from the observers point of view.

The E_k of $B$ viewed from $C$'s point of view = 1,392*10¹⁶ J.
The E_k of $A$ viewed from $C$'s point of view = 6,00*10¹⁶ J.
The E_k of $A$ viewed from $B$'s point of view = 1,392*10¹⁶ J.

Now from $C$'s point of view, $A$ gained 4,606*10¹⁶ J.
And from $B$ point of view, $A$ only gained 1,392*10¹⁶ J.

Now my question is, how do you explain this?I used 3*10⁸ m/s as the speed of light to make things easier.

 

[beamie564]

Is your doubt something like this : The change in velocity for both A and B is the same, yet the change in their kinetic energies are different. How?
If so, the answer is simple. The slope of the curve in $K$ vs $v$ graph is not a straight line. It keeps on increasing (the curve has an asymptote at $v=c$). It means that as the value of $v$ increases, so does the slope. It means for a given $\Delta v$ , $\Delta K$ is greater for larger values of $v$.

 

[beamie564]

The same thing is seen in non relativistic domain also. The change in kinetic energy of a body going from $100 m/s$ to $101 m/s$ is greater than the change in $K$ for a body going from $0 m/s$ to $1m/s$.
The reason is same here. The slope $(\Delta K)/(\Delta v)$ is not a straight line.

 

[Wout Veltman]

Is your doubt something like this : The change in velocity for both A and B is the same, yet the change in their kinetic energies are different. How?
If so, the answer is simple. The slope of the curve in $K$ vs $v$ graph is not a straight line. It keeps on increasing (the curve has an asymptote at $v=c$). It means that as the value of $v$ increases, so does the slope. It means for a given $\Delta v$ , $\Delta K$ is greater for larger values of $v$.

That does come very close to what I ment, yes, thank you very much.