# Total force needed to maintain constant acceleration of a bus

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 ?

When the mass is sliding back relative to the bus there is a "backwards" pseudoforce, $$m_{\rm mass}a$$, on the mass making it accelerate "backwards" relative to an observer travelling with the bus.
When the mass is attached to the back of the bus now exerts a "forward" normal force, $$m_{\rm mass}a$$, on mass equal in magnitude to the "backwards" pseudoforce that the mass experienced when sliding back, so the mass no longer moves relative to an observer on the bus.
As far as the observer on the bus is concerned when observing the bus and mass attached to bus system there is the original "backwards" pseudoforce acting on the bus when the mass was sliding, $$m_{\rm bus}a$$, and the "backwards" pseudoforce on the mass,$$m_{\rm mass}a$$, which together, $$m_{\rm bus}a+m_{\rm mass}a$$, equal in magnitude to the now larger external "forward" force causing the bus and mass to accelerate.
Your given forces equation $$F_{ni}=F_i+F_{pseudo}=Ma-Ma=0$$ is invalid in a non-inertial (bus system) reference frame.
Roller-skater has no any reason to move with acceleration (say nobody in a bus is specifically pushing that person backwards). So the only reason for a person to move backwards is a pseudo force which arises due to bus movement with acceleration itself. Thus equation should be $$F_{roller-skater} = F_{pseudo} = -m_{skater}~a_{bus}$$ Negative sign is because pseudo force acts in opposite direction to bus movement.
Your given zero net force would be true if say roller-skater would have a backpack with rocket-engine, and in the moment bus starts moving, it would fire-on rocket engine and would start propelling forwards with same $$a_{bus}$$. Thus in the end, it would stay at rest in the bus same place, because rocket engine force would compensate pseudo force due to bus movement.