Getting back to the original post here, what you are trying to do is not to prove Newton's second law in the mathematical sense but to merely verify it - to show that it gives a correct prediction of what happens in nature.
This experiment is sometimes used in high school physics to demonstrate the second law, but it has a particular flaw: the tension force on the string pulling the trolley varies not only with the mass of the falling weight but also with mass of the trolley. What you need is the same constant pull on the trolley for different trolley masses. You would need to pull the car with elastics or springs stretched a set amount.
If you do that, you can show that the acceleration is proportional to the number of springs or elastics and inversely proportional to the mass of the trolley, just as the second law predicts. That demonstrates that Newton's second law provides the correct result. And you will not have used Newton's second law in order to produce your data.