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A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

In this problem we are faced with an apparently easy area problem, but it has gone horribly wrong! What happened?

### From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

##### Stage: 4 Challenge Level:

Interpreting graphs is a really important problem-solving skill - well done Sam from Dorset for thinking this through so carefully, and for leaving us with an excellent question at the end.

All the graphs start with a rising straight line. That stage is when the bead falls directly down.
The speed and the vertical velocity will be the same thing.

Next there's a jolt when the bead hits the rail and gets shifted sideways.
On graphs 1 and 3 that's a big loss of vertical velocity.
Graph 2 has less of a jolt and graph 4 doesn't seem to have any jolt.

Now for the bead moving down the rail : If the rail was flat (horizontal) vertical velocity would be zero.
So if the bead had some vertical velocity and the rail flattened out that would show on the graph as vertical velocity falling - graphs 2 and 4 do that.

Graph 2 actually goes to zero

Graph 4 doesn't continue far enough to say for sure

If the rail was a straight line (sloping down) the bead would get faster and faster just like something rolling down a hill, and of course that means the vertical velocity is also increasing.

The rail for graph 1 is nearly a straight slope so the graph shows a fairly steady increasing vertical velocity but not as steep as the straight drop right at the start. Things speed up quickly on a steep hill but only speed up slowly on a gentle slope.

The rail for graph 3 descends more and more steeply so the plot for vertical velocity becomes more and more steep.
I thought of the rail like lots of very short straight rails each having a steeper gradient than the one before.

Now for the main challenge :

For the vertical velocity to stay at a fixed level my reasoning went like this.

I know that a straight line with even the slightest slope makes the vertical velocity increase steadily and I know that the rail flattening out can make the vertical velocity decrease so there must be some shape, just a bit bowed downwards from a straight line that will make the vertical velocity neither increase or decrease but stay exactly the same.

I think I've done quite well but I can't help the jolt at the start.

It seems that I need some flattening out of the rail but not much.

I don't know what sort of curve that is mathematically.
An arc from a circle seems too much curved.
Maybe some kind of big oval ?

That's an excellent question to finish with Sam - seems a curve isn't just a curve!