Internal forces in a system and the cause of change in velocity of a system 
Let's say a cart of mass $\rm M$ is moving with a constant velocity $\rm V$. A block of mass $\rm m$ is dropped into the cart with no horizontal velocity.

Applying conservation of linear momentum in the direction of motion of the cart, we can say that, the new velocity $v'$ of the system (block $+$ cart) is $\mathrm{MV/(m+M)}$. I can also say this using impulse-momentum equations.

My question :
(a) The impulsive forces acting between the block and the back of the cart are internal forces in my cart $+$ block system. So where is the external force that brought about the change in the velocity of the system?
(b) Another question related to internal forces and change in velocity :
If I take the earth and the sun as a system, there is clearly no external force acting here. But the direction of velocity of the centre of mass is changing.(since the earth moves around the sun and the centre of mass lies between them) You could say that this is because there is no external torque and the angular velocity must be constant. But which force provides the centripetal acceleration to the centre of mass, causing it to change velocity?
 A: You can't use conservation laws (like conservation of momentum) but change the system in the middle.
You are looking at the cart with velocity $v$.  But the system after the collision is the cart with mass $M+m$.  So that has to be your system.  The system with both masses does not have an average velocity of $v$.  There is no change in the velocity of the system.  The $v=0$ of mass $m$ has be included.
If you limit your system to just the mass $M$, then it experiences an external force that slows it down.

But the direction of velocity of the centre of mass is changing.(since the earth moves around the sun and the centre of mass lies between them)

No.  If we're ignoring the rest of the solar system, then the center of mass of the sun-earth system does not change velocity over the course of the earth's orbit.  The center of mass lies well inside the sun.  The geometric center of the sun would orbit around that center of mass, but it is very close to the center of the sun (only about 500km away).
A: 1.The velocity of the center of mass does not change, only the velocity of m and M
2. Sun and earth (taken without the other planets) move around their common center of mass which does not rotate, it is inside the sun, so you usually forget this.
A: So where is the external force that brought about the change in the velocity of the system?
What do you mean by "velocity of the system?"
As you use in your calculation, the horizontal momentum of the system is the same before and after the block is dropped as there is no horizontal force external to the system - since the cart moves at constant speed friction is assumed to not be present and gravity acts vertically. The momentum of the cart and the block individually change and this can be explained by the internal forces acting on each object. For instance, if you take the block to be the system, then the cart exerts a horizontal force on the block that changes the block's momentum in that direction.
