# I Speed and kinetic energy in different inertial frames.

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1. Oct 26, 2016

### Wout Veltman

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

From Bob's perspective Chris is travelling with 1.5*108 m/s in direction b (The opposite of x). Angelica is travelling 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 Ek of Bob. I also calculated the Ek of Angelica, all from chris' perspective. Now the outcome that I was expecting was that Bob's Ek would be half of Angelica's Ek when looking from Chris' perspective. Because Angelica is also travelling 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.

Last edited: Oct 26, 2016
2. Oct 26, 2016

### Aniruddha@94

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.

Last edited: Oct 26, 2016
3. Oct 26, 2016

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

4. Oct 26, 2016

### Aniruddha@94

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 $v$.

5. Oct 26, 2016

### Aniruddha@94

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.

6. Oct 26, 2016

### Wout Veltman

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

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