Does acceleration follow the same addition like velocity? Two frames have relative velocity $u$, but an object in one is initially at rest and in the other has velocity $v$. If it then starts to accelerate, the acceleration measured in both frames is equal.
But, if one frame (S) is accelerating at $5\text{ m/s}^2$ and the other (S') at $3\text{ m/s}^2$ in the same direction. As viewed from S' would you see S accelerating at $2\text{ m/s}^2$? 
 A: Yes. One way to see this: if $S'$ and $S$ have coordinates $x'$ and $x$, then by the usual rule we know that $S'$ observes a distance of $\Delta x = x-x'$ between them. Differentiating on both sides, we get $\Delta v = v - v'$, $\Delta a = a - a'$, and so on. In other words, velocity, acceleration, and all higher derivatives behave like you think they should.
The key step here was being able to differentiate on both sides; this fails in special relativity because observers won't agree on $dt$.
A: Yes, you're correct. Since $S'$ is an non-inertial/accelerated frame of reference, all objects within this frame is acted upon by a pseudo force that is proportional to the mass of the object and whose direction is opposite to the direction of acceleration of $S'$. The fact that $S$ is accelerating(with respect to a rest frame on ground) with $a=5 m/s^2$ implies that every object with mass $m$ in $S$ is acted upon by a force $F=ma$. In $S'$ frame two forces act on the same object,the force $F$ in the direction of acceleration of $S$, and the pseudo force $F_\text{pseudo}$ whose direction is opposite to the acceleration of $S'$(and $S$ as well), therefore it's negative.  so using Newton's second law to calculate the acceleration  as measured by $S'$ which I call $a_\text{in s'}$:
$F-F_\text{pseudo}=ma_\text{in s'}$
Since pseudo force is proportional to the object's mass so $a_\text{pseudo}=3$  and $F=ma$ 
Where $a=5$, therefore:
$ma-ma_\text{pseudo}=ma_\text{in s'}$. dividing by $m$ and substituting one gets:
$5-3=2=a_\text{in s'}$. 
Just as you asserted. 
A: Yes, as viewed from S', S accelerates at 2 m/s2. One way to think of this is just imagining S' as a rest frame, meaning that the rest frame is travelling with an acceleration of 3m/s2 through 'absolute space'. Alternatively, just comparing the velocities of both frames after each second confirms that they are moving apart with an acceleration of 2m/s2.
