Irodov Problem 1.15

a) Relative to elevator the acceleration of the bolt is 9.87 + 1.2 = 11.07 m/s². At the moment the bolt started to fall, its velocity relative to the elevator was 0. Thus, the time taken by the bolt to reach the elevator floor is




b) Displacement:
Let 0 be the position of the elevator ceiling when the bolt starts to fall. The elevator floor at this time is at -2.7 m. The upward velocity of the elevator at this point is 1.2 x 2 = 2.4 m/s. After 0.7 seconds the position of the elevator will be at
-2.7 + 2.4(0.7) + 1 2 (1.2) (0.7)² = -0.726 m.
Since the bolt essentially traveled from ceiling to floor, its final displacement was same as the position of the floor. Thus, the bolt displaced about 0.7 m.

Distance:
The reason distance is different from displacement is because the bolt will first travel upwards before it starts to descend. Distance calculation must hence include the length of the entire path followed by the bolt. The bolt will travel for 2.4/9.8 = 0.243 sec and is at a height of
2.4 x (0.243) – ½(9.87)(0.243)² = 0.292 m
from the point it detached from the elevator before it begins to descend towards the elevator floor. For the rest of the time 0.7-0.243 = 0.457 sec the bolt free falls starting at 0 velocity.
The distance traveled during this time is given by,
½(9.87)(0.457)² = 1.03m.
The total distance traveled by the bolt is thus given by 0.292 + 1.03 = 1.322 m.