An elevator is moving upward at a constant speed of 2.50 m/s. A bolt in the elevator ceiling 3.00 m above the elevator floor works loose and falls. (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor (b) according to an observer in the elevator? (c) According to an observer standing on one of the floor landings of the building? (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

a) t = 0.782

b) v_b/el = - 7.67 m/s

c) v_b/e = - 5.17 m/s

d) y_b/e = 1.04 m down-wads

Explanation:

Given:

- initial position of bolt y_b, i = 3.0 m

- initial velocity of bolt v_b, i = 2.50 m/s

- constant velocity of elevator v_e = 2.50 m/s

- acceleration of free fall for bolt a = 9.81 m/s^2

Find

- (a) How long does it take for the bolt to fall to the elevator floor? What is the speed of the bolt just as it hits the elevator floor

- (b) according to an observer in the elevator?

- (c) According to an observer standing on one of the floor landings of the building?

- (d) According to the observer in part (c), what distance did the bolt travel between the ceiling and the floor of the elevator?

Solution:

- The position of bolt y_b is given by kinematic equation of motion:

y_b = 3.0 + 2.5*t + 0.5*9.81*t^2

- position of floor with constant upward speed is:

y_f = 2.50*t

- When bolt hits the floor they have same position:

3.0 + 2.5*t + 0.5*9.81*t^2 = 2.50*t

4.905*t^2 = 3.0

t = sqrt (3/4.905) = 0.782 s

- velocity of the bolt relative to earth is:

v_b/e = 2.50 - 9.81*0.782

v_b/e = - 5.17 m/s

- Hence, relative to observer in elevator:

v_b/el = - 5.17 - 2.5

v_b/el = - 7.67 m/s

- Relative to earth the distance traveled by bolt is:

y_b/e = 2.5 - 0.5*9.81 * (0.782) ^2

y_b/e = - 1.04 m (downwards)