To find the time taken for the cage to reach the top, we need to break down the different phases of the motion.
Phase 1: Acceleration
First, we need to calculate the time taken during the acceleration phase. We can use the equation:
v = u + at
where:
v = final velocity = 9 m/s
u = initial velocity = 0 m/s
a = acceleration = 1.5 m/s^2
t = time taken during acceleration phase
Rearranging the equation, we have:
t = (v - u) / a
t = (9 - 0) / 1.5
t = 6 seconds
Phase 2: Constant velocity
During this phase, the velocity remains constant at 9 m/s. To find the time taken during this phase, we can use the equation:
t = distance / velocity
In this case, the distance is equal to the height of the cage, which is 120 m. Thus:
t = 120 / 9
t ≈ 13.33 seconds
Phase 3: Retardation
In this final phase, the cage experiences retardation with an acceleration of -6 m/s^2. The initial velocity is 9 m/s, and the final velocity is 0 m/s. We can use the equation:
v = u + at
where:
v = final velocity = 0 m/s
u = initial velocity = 9 m/s
a = acceleration = -6 m/s^2
t = time taken during retardation phase
Rearranging the equation, we have:
t = (v - u) / a
t = (0 - 9) / -6
t = 1.5 seconds
Total time taken:
To find the total time taken for the cage to reach the top, we sum up the times taken during each phase:
Total time = time during acceleration phase + time during constant velocity phase + time during retardation phase
Total time = 6 seconds + 13.33 seconds + 1.5 seconds
Total time ≈ 20.83 seconds
Therefore, the cage takes approximately 20.83 seconds to reach the top.