Answer to Question #146995 in Calculus for Sean

Question

In an opera house, the base of a chandelier is 160 ft above the floor. Suppose the Phantom of
the Opera dislocates the chandelier and causes it to fall from rest and crash on the floor.
a. What is the equation of motion of the chandelier?
b. Find the instantaneous velocity of the chandelier at 1 sec and 1.5 sec.
c. Find how long it takes the chandelier to hit the floor.
d. What is the speed of the chandelier when it hits the floor?

Answer 1

a. To begin wtih, if chandelier goes down from the rest, it`s motion is considered as falling freely. This means that it falls with g = 9.81 m/s<sup>2</sup>=32.18 ft/s<sup>2</sup> and it`s height can be computed by:

h=g*t²/2 but we have given maximum height h_max=160 ft.

b. To find velocity in given time, we need to find the height (height that it falls in given time) first and calculate velocity:

h(1sec)=g*(1)²/2=16.09 ft and

h(1.5sec)=g*(1.5)²/2=36.205 ft from the rest.

we can calculate velocity by h=(V²_final-V²_initial)/(2*g)

Initial velocity is V_initial=0, Then:

V²_final=2*g*h and V_final=$\sqrt{\smash[b]{2*g*h}}$ .

If we put values of unknowns:

V(1sec)_final=$\sqrt{\smash[b]{2*g*h}}$(1sec) =32.18 ft/sec and

V(1.5sec)_final=$\sqrt{\smash[b]{2*g*h}}$(1.5sec) =48.271 ft/sec.

They are answers!

c. We can answer the third question by:

h_max=g*t²/2 --->t=$\sqrt{\smash[b]{2*h(max)/g}}$ =3.15sec, so 3.15 sec is enough to fall floor fully for chandelier.

d. We can find the speed of chandelier when it reachs floor by:

h_max=(V²_final-V²_initial)/(2*g), V_initial=0 then:

V²_final=2*g*h and V_final=$\sqrt{\smash[b]{2*g*h}}$(max)=101.47 ft/sec.

Chandelier reachs 101.47 ft/sec when it falls floor fully.