Finding the coordinates of a point in time, given start(x,y) and end(x,y)

Question

For a little programming project that should animate a billiards ball (without friction and in a vacuum), I need to find out the x and y coordinates of the ball at any point in time given its start and end coordinates and its speed.

Let's say, the ball starts from coordinate x = 10.1 cm, y = 10.5 cm and will hit the hole at coordinate x = 500 cm, y = 200 cm. The speed of the ball is 7.0 cm per second.

I want to show the ball at any point in time, where time starts at 0 seconds.

So far I thought of using Pythagoras somehow:

x^2 + y^2 = z^2

where x and y are the current coordinates and z being the distance that the ball rolled from its start location up to the current point in time.

Since I know that the total x distance is 489.9 cm and the total y distance is 189.5 cm, I assume that I can write y as a factor of x, i.e. 189.5 / 489.9 which results to y_factor = 0.3868....

Now I need to figure out the x and y coordinates at some point in time, say 3.5 seconds.

According to my Pythagoras thought: x^2 + (0.3868*x)^2 = 3.5

I think that leaves me the following:

distance =  speed * time_passed
current_x = squareroot(distance) / squareroot(y_factor^2 + 1)
current_y = current_x * y_factor;

To my astonishment, the animation gets slower and slower over time and I have no idea why.

Where am I wrong here?

As you can clearly see, my understanding of mathematics is more than dusty, so please try to write in laymans terms (or programming jargon)..

Answer 1

Here, $x$ and $y$ are the distances traveled horizontally and vertically rather than the actual coordinates. These are an offset you should apply to the initial coordinates.

You establish that $x^2+y^2=d^2$. You have $y=fx$, where $f$ is your factor between $x$ and $y$. This gives $$x^2+(fx)^2=d^2$$ $$x^2+f^2x^2=d^2$$ $$x^2(1+f^2)=d^2$$ $$x^2=\frac{d^2}{1+f^2}$$ $$x=\left(\frac{d^2}{1+f^2}\right)^{0.5}=\frac{d}{(1+f^2)^{0.5}}$$

You shouldn't be taking the square root of your distance in your second line of code.

A consideration you could make is that at any point this is essentially one expanding triangle with the same angles. So $$\frac{d_{final}}{d_{current}}=\frac{x_{final}}{x_{current}}$$ as per similar triangles, and you know $3$ of these. Replace $x$ with $y$ to get that instead. Saves you from square roots, which are more computationally expensive.

Answer 2

The simplest way I see is to parameterize x and y in terms of t. Which you can do using the following.

$$x(t)=v_{x}t+x_{0}\\y(t)=v_{y}t+y_{0}$$

Where $x_{0}$ and $y_{0}$ are the starting positions. So now we just need to solve for $v_{x}$ and $v_{y}$.

$$v_{x}=v\cos{\theta}=v\cos{(\tan^{-1}\frac{\Delta y}{\Delta x})}=v\frac{\Delta x}{\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}}\\v_{x}=v\sin{\theta}=v\sin{(\tan^{-1}\frac{\Delta y}{\Delta x})}=v\frac{\Delta y}{\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}}$$

Where $v$ is the speed, $\Delta x$ is $x_{final}-x_{initial}$, and $\Delta y$ is $y_{final}-y_{initial}$.

I hope this is simple enough. I can try to write some pseudocode if you're having trouble understanding.

Answer 3

$x^2 + (0.3868 \times x)^2 = 3.5$ is an equality between a distance (squared) and a time... It doesn't work, at a very fundamental dimensional level.

Using units (or at least dimension) is a must to reason properly about a physical question. Here you have 3.5s, so you want to convert this into a distance. Since you have a speed, it behoove you to use $v = \frac{d}{t}$ or more exactly here $d = v \cdot t$ so in 3.5s you'll go $7 \times 3.5$ cm. That's only a distance (the hypotenuse) so you should square it in your precedent equation : $$x^2 + (0.3868 \times x)^2 = (7 \times 3.5)^2$$

Let's reformulate with letters (names are overly long and obscure the computation, we're not programming yet) : $$x + (f \cdot x)^2 = (v \cdot t)^2$$ $$(1 + f^2)x^2 = (v \cdot t)^2$$ $$x^2 = \frac{(v \cdot t)^2}{1 + f^2} $$ $$x = \sqrt{ \frac{(v \cdot t)^2}{1 + f^2} } $$ $$x = \frac{v \cdot t}{\sqrt{1 + f^2}}$$

So the problem was that you took the square root of $v \cdot t$.

Answer 4

The point is that the formula for current x is given by the distance (not the square root of the distance) divided by the quantity (y_factor^2+1).

In fact, starting from the Pythagorean equation x^2 + (0.3868*x)^2 = distance^2, which expresses the relation between the distance that the ball rolls and the change in the x-coordinate, we get:

x^2 *(0.3868^2 + 1) = distance^2

x^2 = distance^2/(0.3868^2 + 1)

x = distance/sqr(0.3868^2 + 1)

The animation based on this formula would probaby be realistic and no longer slow.

Answer 5

$\left({ {10.1+7t\cos\theta} \mid {10.5+7t\sin\theta} }\right)$,

where $t=$ time in seconds and
$\theta=tan^{-1}{{200-10.5}\over{500-10.1}}$.