Find the acceleration of the specified object.

Question

Find the acceleration of the specified object. (Hint: Recall that if a variable is changing at a constant rate, its acceleration is zero.)

A ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. Find the acceleration (in ft/sec²) of the top of the ladder when the base of the ladder is 15 feet from the wall.

__ft/sec²

Answer 1

ladder is 25 feet, resting against a wall

when the bottom end is 15 feet away from the bottom of the wall, moving at a rate of 2 feet per second, how fast is the top of the ladder moving down the wall and how much is it accelerating?

this is a related rates problem, involving derivatives

h^2 = b^2 +a^2

25^2 = 15^2 + a^2

a^2 = 625-225 = 400

a = 20 feet high

take derivatives of the Pythagorean Theorem equation

2hh' = 2bb' + 2aa'

divide by 2

solve for a''

a' = (hh' - bb')/a

plug in the values

a' = ( +25(0) -15(2))/20

= -30/20

= -3/2 feet per second

=-1.5 ft/sec = velocity of the tip of the ladder

a" = ((-bb')/a)'= -.3125 ft/sec^2= acceleration of the tip of the ladder

assuming b"=0

otherwise

a" = -3125 -.75b"