34. According to the model we used to solve Example 2, what happens as the top of the ladder approaches the ground? Is the model appropriate for small values of $y ?$
To find: Find the validity of the model when top of the ladder approaches to the ground.
The model not valid when top of the ladder approaches to the ground.
A ladder which is rests against a vertical wall and the on the horizontal ground.
Suddenly the bottom of the ladder slides away from the ground at a rate of 1 ft/s.
(1) Chain rule:
(2) Pythagorean Theorem.
Let us assume that be the length of the ladder and be the horizontal distance from the wall to the lower end of the ladder, and be the vertical distance from the ground to the top end of the ladder as shown in the Figure 1 given below.
Since the ladder slides away from the wall so the distance become increases and decreases.
Therefore and are function of the time
Now from the Pythagorean Theorem.
Differentiate with respect to the time .
On further simplification
As top of the ladder approaches to the ground, that is .
Taking limit of as .
Therefore, as the value of which does not make physical sense.
For example, the model predicts that for sufficiently small ,the top of the ladder start moving speed of greater than the speed of light, but according to the special theory of relativity ,nothing can move faster than light. Therefore the model is not appropriate for the sufficiently small value of .
Due to the reason which is actually happen that the ladder leave the wall at some point in its decent.
Therefore, the model not valid when top of the ladder approaches to the ground.