Now let's take as an example $x(t)=bt^2$. Secondly this derivative is only defined if $x'(t)\neq 0$ so there are some constraints in doing this. These are simple measurable quantities of distance, angle or something else geometrical forming a generalized coordinate vector $$\boldsymbol$ because we can write $v$ as a function of $t$ and we can also write $t$ as a function of $x$. First, we decide what the degrees of freedom are and assign generalized coordinates to them. Looking at a complex mechanical system, like a human launching a ball while riding on a skateboard. The common term for these "pseudo-independent" quantities is generalized coordinates. They are variables humans bless as independent in order to answer what-if scenarios and to establish mathematical models of systems byways of separation of variables. It is true, that in nature there is only one true independent variable, time. You should have in mind, that at least in theory, we always check our calculations with experiment, so we have an experimental proof instead of a mathematical one (generally). If there are special cases, they could be "obviously strange". I am not sure if there is a special case or not, but for physicists it is not important, because in 99.9% this will be true. From this it follows, that $v$ can be rewritten as function of either $t$ or $x$. This implys, that $dx/dt$ has allways that some non infinite value. And furthermore, we don't have infinite speed in real life. In order to define velocity, the object has to change its position in some amount of time. The thing is, that almost all functions, which can appear in nature or real life systems are, in most times, continuous and differentiable. Because we don't bother to cancel out derivatives, and we "NEVER" check if we can imply some rule on our equations. Well, this is the most common thing for which mathematicians make fun of physicists. Many books take it for granted that you're being careful about this, so they don't have to worry about it on your behalf. You also have to make sure you don't try to pull this stunt when the ball is at the very top of its trajectory (or more generally, at points where its velocity is zero). If you just write velocity as a function of height, you do have to be careful to make it clear from context which of the two functions - the "on the way up" function and the "on the way down" function - you're referring to. All of this is part of the content of the implicit function theorem, which you can google for. And moreover that function is differentiable and obeys the chain rule. So velocity is definitely not a (global) function of distance.īut this much is true: For any height $h$ except for the maximum height the ball ever reaches, there is some open interval around $h$ - some range of heights from $h-\epsilon$ to $h \epsilon$ - in which you can treat velocity as a well-defined function of height while the ball is on its way up, and another well-defined function of height while the ball is on its way back down. When it is at the same height $h$ on the way down, it has a negative (downward directed) velocity. When the ball is at height $h$ on the way up, it has a positive (upward directed) velocity. For example, as one commenter has already mentioned, throw a ball directly up in the air and wait for it to come down. You are correct that you cannot (globally) write velocity as a function of distance.
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