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You will see that this angle will be less than $90^{\circ}$ for a relativistic impact. … I was thinking of maybe using a new reference frame $S'$ where the momentum of each state is zero, and somehow getting the final velocity using velocity addition formulas maybe? …

In addition, there is a magnetic field that circulates around the line of travel of the charge, and this magnetic field is proportional to the electric field (i.e., stronger at points perpendicular to … One does, however, have to be careful of what one means by $\mathbf{p}$ here; it has to be defined as the relativistic momentum $\mathbf{p} = \gamma m \mathbf{v}$ for this equation to hold. …

the velocity of the particle after the collision. … We could do more involved calculations involving the Lorentz transformation of momentum and relativistic velocity addition etc. but we still end up with the same result. Gas pressure is invariant. …

If they exist, they have to obey the relativistic velocity addition rule (using units such that c=1): $$ u = \frac{v + u'}{1 + vu'} \ ,$$ just like any other particle: If an observer S' is moving with … velocity v relative to observer S sees a particle moving with velocity u' then the observer S will see the particle moving with velocity u. …

What you are not taking into account is the relativistic velocity addition. … To calculate the velocity of $\text{A}$ from $\text{B}$'s point of view ($v_{\text{AB}'} $) we use the relativistic velocity addition formula. …

A relativistic fly flies at 0.7c in the same direction as a car traveling at 0.8c. According to the driver of the car, how quickly will the fly approach the car? … To solve this problem, I applied the relativistic velocity addition formula: $$ u' = \frac{u - v}{1 - \frac{uv}{c^2}} $$ where u = 0.7c and v = 0.8c. …

If $\text{A}$ decided to fire a relativistic gun towards the front of the train that fires bullets at $0.9c$ in the rest frame of the gun, $B$ would measure the bullet to be moving slowly away from $\text … The velocity of the bullet relative to $\text{B}$ is just under $0.99c$. This is a property of relativistic velocity addition. …

The recession rates associated with Hubble's law are not velocities in the relativistic sense (i.e., angles in spacetime), so there's no reason they should be limited to the speed of light. … Notice for example that they are linearly additive, whereas velocities in relativity add nonlinearly according to the relativistic velocity addition formula. …

velocity addition formula : $$v_s = \frac{v +u'}{1+vu'}\ = \ \frac{0.5 +0.1}{1+0.5\times 0.1} \approx 0.57142,$$ where $u'$ is the sound velocity measured by Ben in his rod at rest in his rocket and … You can prove this to yourself by entering $v' = 1$ in the velocity addition equation above. The laws of physics are the same for both you and Ben. …

Now we boost that to the moving frame so that $B$ is stationary: $$ v'_B = 0 $$ We need to use the relativistic velocity addition to get: $$ v'_A = \frac{2v}{1+v^2} \lt 2v $$ The initial momentum is ($ … the $AB$ lump: $$ p'_f = \gamma Mv $$ $$ p'_f = \gamma(2m\gamma)v$$ $$p'_f = 2\gamma^2 mv $$ so you need $E=mc^2$ and relativistic velocity addition to conserve momentum: $$p'_f = p'_i$$ very cool. …

Here I'm using some kind of Galilean transformation for temperature so that $T_B^A = T_B-T_A$ (just like velocity addition). … in relativistic kinematics. …

It is well known that there are few kinds of redshifts (e.g. relativistic, cosmological, gravitational). … Since the proper velocity in cosmology $v$ is addition of Hubble flow and peculiar velocity, i.e. $v=v_\mathrm{pec}+Hd$, where the $d$ is proper distance, I don't know what to put into the formula. …

with non-relativistic velocity) the difference is negligable. … All entities that can be represented with a vector (position, velocity, acceleration) add and subtract according to the rules of vector addition in euclidean space. …

Moreover, together with the relevant rule of velocity addition, it even determines the exact form of the kinetic energy as a scalar function of the speed of motion in either context." … The author of the above article proves that the change in kinetic energy of an isolated system is independent of the observer's velocity. …

However, velocities in relativity add in a special way; see the relativistic velocity addition formula. … The cosmological recession rate is computed by instead just adding the relative velocities naively, without using relativistic velocity addition. …