Question

An elevator moves downward in a tall building at a constant speed of 5.05 m/s. Exactly 5.75 s after the top of the elevator car passes a bolt loosely attached to the wall of the elevator shaft, the bolt falls from rest.

(a) At what time does the bolt hit the top of the still-descending elevator? (Assume the bolt is dropped at t = 0 s.)


(b) Estimate the highest floor from which the bolt can fall if the elevator reaches the ground floor before the bolt hits the top of the elevator. (Assume 1 floor ≌ 3 m.)

Answer 1

The problem calculates the time for a bolt to strike a descending elevator and estimates the highest starting floor. It involves equations of motion and accounts for the constant speed of the elevator and the acceleration due to gravity on the bolt.

Step-by-step explanation:

The physics problem involves calculating the time it takes for a bolt to hit the top of a descending elevator and estimating the highest floor from which the bolt can fall without hitting the ground floor first.

(a) Time for the bolt to hit the top of the elevator

The bolt starts falling after the elevator has been descending for 5.75 seconds at a steady speed of 5.05 m/s. By this time, the elevator has descended 5.75 s × 5.05 m/s = 29.0375 m. As the bolt falls from rest, it will accelerate due to gravity, which is approximately 9.81 m/s^2. Therefore, the time it takes for the bolt to hit the top of the elevator can be found using the equation of motion s = ut + 0.5at^2, where s is the distance fallen, u is the initial velocity (0 m/s in this case since the bolt falls from rest), a is the acceleration due to gravity, and t is the time elapsed.

Setting the distance s to 29.0375 m and solving the quadratic equation for time, we find a value for t that represents how long it takes for the bolt to fall the distance the elevator has traveled since the bolt started falling. This value of t must be added to 5.75 seconds to determine the total time from the moment the elevator started descending.

(b) Estimating the highest floor

To estimate the highest floor from which the bolt can fall before the elevator reaches the ground floor, we assume that one floor is approximately 3 m in height. By dividing the total distance the elevator descends before the bolt hits its top by 3 m/floor, we can get an estimate of the number of floors. This assumes the elevator continues to move at a constant speed until the bolt makes contact.

Answer 2

Step-by-step explanation:

The elevator is moving with uniform speed and bolt will fall with acceleration due to gravity g .

Displacement of elevator when bolt starts falling

= 5.05 x 5.75 = 29 m

Let after time t bolt reaches the elevator

in time t , displacement of bolt = 1/2 g t²

= .5 x 9.8 x t² = 4.9 t²

displacement of elevator = 5.05 t

position of elevator = 29 + 5.05 t

According to question ,

29 + 5.05 t = 4.9 t²

4.9 t² - 5.05 t - 29 = 0

t = 3 s approx .

b )

Distance covered by the bolt

= 1 /2 g t²

= .5 x 9.8 x 9

= 44.1 m

Bolt must have to fall from a height of 44.1 m

or in terms of floor it is

44.1 / 3 = 15 th floor .