Suppose there is a person wearing roller-skates, inside a bus ( to neglect the friction on the floor ). The mass of this person is $m$ and the mass of the bus $M$.
Suppose, the bus now starts to accelerate at an acceleration $a$. The mass is at rest relative to the bus, so, in the non-inertial frame of the bus, the mass tends to move backward and sticks to the wall of the bus.
Before the mass sticks to the wall of the bus, let us look at the forces on the bus, in the non-inertial frame.
$$F_{ni}=F_i+F_{pseudo}=Ma-Ma=0$$
As we know, pseudoforce is directly equal to negative of acceleration times the mass of the object.
Now suppose, the mass sticks to the bus. Since the bus is moving with the same acceleration, in the non-inertial frame, it should appear to be at rest, and so $F_{ni}=0$ still. However, now there is an added force, the normal reaction due to the mass sticking on the back surface.
If I want to maintain the total acceleration, shouldn't the Force on the bus be increased ?
The initial force on the bus was $Ma$, but now, shouldn't it become $(M+m)a$ ?
In that case, the pseudoforce on the bus is still $-Ma$, and the normal reaction of the man is $-ma$. Hence the total force again comes out to be $0$.
Thus, in order to preserve the total acceleration of the bus to be constant, the force must be increased as soon as the man collides with the back wall, right ? In a sense, the mass of the system has increased. Is this the correct reasoning ?