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By studying mathematical objects themselves (such as derivatives) which we use to model some natural phenomenon (such as gravity), we can often learn more about the physical phenomenon by playing around with the mathematical model. Mathematical models allow us to ask the 'what if we change this' question.
Mathematics greatly enhances our ability to study things empirically, giving us insight into nature. And it simplifies a myriad number of facts.
The study of calculus is a good example of this, we study calculus for its own sake so that later we can use it to study physical phenomenon (and then we can play with the model and learn more about nature).
However we must always check that the physical reality agrees with the mathematical model when we change our initial conditions or assumptions in the model. Sometimes you can get unexpected results when you consider negative time or negative height (when working on a trajectory problem).
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