Velocity-Time graph confusion

Question

I'm not 100% sure whether this is best answered with regards to Maths or Physics, but I feel this is more mathematical.

If you take the equation $y = 40-10x$. If you change $y$ to $v$ (velocity in $ms^{-1}$), and $x$ to $t$ (time in $s$), you can use this to model the velocity of a ball, thrown upwards at $40ms^{-1}$ (to keep things simple, I am assuming the acceleration due to gravity is $10ms^{-2}$ and ignoring air resistance).

Now, if I make $v=0$, then $0=40-10t \rightarrow 10t=40 \rightarrow t=4$. So, at exactly 4 seconds, the velocity is $0$. This makes total sense to me.

However, if I rearrange the equation to be in terms of $t$, I get $10t=40-v \rightarrow t=4-\frac{v}{10}$. Now, if I make $t$ equal to $4+h$, I get $4+h=4-\frac{v}{10} \rightarrow h=\frac{-v}{10} \rightarrow v=-10h$. As you can see, as long as h isn't $0$, regardless of how small $h$ is, if $t=4+h$, $v$ will not be $0$.

If we go back and remember that this is a ball that has been thrown upwards, then obviously it must "stop" shortly before falling back down. However, as I have shown, it stops for absolutely no time at all.

What is confusing me, is that both of the below statements are true;

1. The ball has a velocity of $0$ for no time at all, and if an action takes up $0$ time, then surely this action never happened. However;
2. As I say above, we can clearly prove that the velocity must have been $0$ at one point given it changes direction.

Surely these 2 statements contradict each other. Can anyone explain why it is possible for both to be true?

Answer 1

The result is consistent indeed

• when $h=0 \implies v=0$

• when $h>0 \implies v<0$ since the ball is falling down

Answer 2

"... if an action takes up $0$ time, then surely this action never happened ..."

This is where your error lies. In classical physics we assume that space and time are continuous and can be divided into parts that are as small as we like. Under these assumptions it is possible for an event to have zero duration i.e. to occur at a single instant in time. It is also possible for an event to have zero spatial extent i.e. to occur at a single point in space.

If you have philosophical doubts about these assumptions then you may be reassured by the fact that we know that classical physics is only an approximation to reality. In quantum physics, Heisenberg's uncertainty principle prevents an event having zero duration or zero extension - there is always some unavoidable uncertainty over the time or location at which an event occurs.

But at the scale of throwing a ball, quantum physics has a negligible effect and so we use the methods of classical physics.