The Falling Ladder

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The Falling Ladder
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Symbolic computations on this page use the GeoGebra CAS engine.

What does it feel like at the top of a ladder as the bottom starts to slide away?
If the bottom slides at a steady rate, do you also fall steadily?
If not, then when do you fall fastest?
Can you see that TWO models are actually needed? One for when the ladder is overhanging the top of the wall, and a second when it passes the top of the wall and begins to fall straight down.
If you were at the top of the ladder and the base began to slide away from the wall, at what point would you be falling fastest?
In this model, the ladder is 6 metres in length, and the wall is 4 metres high.
Move the point X to explore this model and to answer questions 1 and 2 below.

x
	0	3.99	6
f(x)
g(x)
h(x)
	4.48
	4.25
	0.35
ladder
wall

App generated by Geometry Expressions

When x = 4.00m, the height (height1) of the top of the ladder is 4.48 m and the length of ladder above the wall (overhang) is 0.350m.

Use the MathBox above to enter a function for graphing, table of values OR expressions for CAS: Define functions, simplify expressions, solve equations...
For example, enter Solve(x^2-x-1=0) and press the CAS button. More?

Hint: When entering mathematical expressions in the math boxes below, use the space key to step out of fractions, powers, etc. On Android, begin entry by pressing Enter.

Type simple mathematical expressions and equations as you would normally enter these: for example, "x^2[space]-4x+3", and "2/3[space]". For more interesting elements, use Latex notation (prefix commands such as "sqrt" and "nthroot3" with a backslash (\)): for example: "\sqrt(2)[space][space]". More?

1. If the ladder is leaning AGAINST the wall (no overhang), then set the point X above to find the height of the top of the ladder when x = 5?

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2. If the ladder is overhanging the top of a wall of height 4 m, then set the point X to find the height of the top of the ladder above the ground when x = 3?

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3. Calculate the length of the overhang from the top of the wall to the top of the ladder when x = 3? (Give numeric answer only)

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4. We will begin by building an algebraic model for the height of the top of the ladder from the ground as the ladder slides down the wall - no overhang. (Before you enter your function, you might go back to the model above and see how well it describes the movement of point P). height1(x) =

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5. Next, build an algebraic model for the length of the overhang: overhang(x) =

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6. You might use similar triangles to define the height of the top of the ladder overhanging the wall with respect to x. (Before you enter your function, go back to the model above and see how well it describes the movement of point P). height2(x) =

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7. Study the trace plots for the falling ladder above. If you are on top of the ladder, then you are falling fastest when x =

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Go back to the graph above and set the point X - does this verify your results? How difficult did you find this activity? Please assign a rating value from 1 (very easy) to 5 (extremely difficult). Comments? Suggestions? What did you learn from this activity?

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The Falling Ladder

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