[fzltrp]

How would you write a program which compute the speed of the ball in your example, and how would you ensure or prove that your program is correct? That's what lambda calculus and all its derived works give you. Others have described it more elegantly here, so please refer to their posts, but if you want references, check out the Curry-Howard isomorphism, Hindley-Milner type system, System F, or what Coq, Agda and similar languages bring on the table.

You could also look at the answers given in the linked stackexchange page, of course.

[Confusion]

Your answer is useless to the commenter you are responding to. If he would 'get' lambda calculus by studying the Curry-Howard isomorphism or the Hindley-Milner type system, he already would have. It is like showing someone that asks about rocket science a calculus book or a rocket and saying: just study how it works. For many people that is completely backwards.

First you need to teach them what the lambda calculus fundamentally is and why it is useful in small bits. Then you can explain to them how powerful it gets, by explaining how the usefulness is extended by the Curry-Howard isomorphism and how you can build the Hindley-Milner type system with it.

The commenter is an 'engineer' and you're a 'theoretician'. As a fellow 'engineer', I completely understand his exasperation with all this pointing to either nice theoretical results or the large scale end products of the theory. That's just fundamentally not how we learn.

[fzltrp]

I am not explaining how, I am merely pointing out that there is a "how to compute" question, and that lambda calculus and type theory help address that question formally. That point tries to address the following question by OP:

> What is the equivalent computer science problem that lambda calculus can help you solve?

That may sound very obvious in retrospect - surely OP knew already what I was saying, and he actually wanted explanations on the how itself. Perhaps that's why it seems useless: I admit that I tend to stick to strict semantics on technical questions, and often completely miss the actual intent. Sorry about that.