Experimental evidence for $F=\frac{dp}{dt}$ Newton's 2nd Law says $\vec{F}= \frac{d\vec{p}}{dt}$ .
In a constant mass system this becomes $\vec{F}=m\vec{a}$. Answers to why $F=ma$ can be found here.
However, in a variable mass sytem (see derivation) Newton's 2nd law still applies and it becomes ${\mathbf  {F}}_{{{\mathrm  {ext}}}}+{\mathbf  {v}}_{{{\mathrm  {rel}}}}{\frac  {{\mathrm  {d}}m}{{\mathrm  {d}}t}}=m{{\mathrm  {d}}{\mathbf  v} \over {\mathrm  {d}}t}$
. What experimental evidence is there for this (both that Newton's law will apply and/or that this equation is correct)?
 A: The Rocket Equation makes use of this variable mass relationship since rockets change weight as they use propellant. So every rocket is experimental evidence of this relationship.
A: Classical physics is not just a set of independent ideas, it is a set of connected ideas, and consequently almost any process involving mass moving from one system to another will offer evidence of the correctness of the 2nd law as it applies to such systems. Simple examples include collisions, rockets, and moving platforms (e.g. a cart, a boat, a car, a train) where some mass falls on the platform or is thrown out. You could also consider a comet, a meteor, a leaky bucket, etc. One never gets complete proof in the sense of a logical deduction, but one gets evidence that the set of ideas is correctly describing the phenomena, and one builds on those ideas.
A: Another version of this is where the mass changes under a scenario of no external force.
There is a classic problem of rain falling into the bed of truck, where one ignores the friction between the truck and the road.
If you have an airtrack, you could do the experiment where you a car slowly slides under its own inertia (F = 0) with a cup attached to the top of the car.  You then incrementally add water from a pipet.  If you define the system to be the moving car and cup, which doesn't experience an external forward force (but not the falling water), then when the water enters the car, the mass increases but at the expense of the velocity.
Alternatively, you can define your system to be the car and falling water.  None of these experience a sideways force.  Here the mass remains constant, though initially the masses are moving at different horizontal velocities.  Horizontal velocity of the car > 0, Horizontal velocity of the falling water = 0.
