A crane is used to lift air conditioning units to the top of a 5-story building. Each story is 3.5 m high. The crane accelerates the units while lifting the units past the first story and then lifts them at a constant velocity past the second through fourth stories. it hen slows to a stop by the top of the fifth story. if the rate of acceleration provided by the crane, both speeding up and slowing down, is 3 m/s^2, how long does it take an air conditioning unit to reach the top of the fifth story.
To find the time it takes for an air conditioning unit to reach the top of the fifth story, we need to break down the process into different stages:
1. Calculate the time taken to accelerate from rest to the maximum velocity.
2. Calculate the time taken to travel at a constant velocity through the second to fourth stories.
3. Calculate the time taken to decelerate from the maximum velocity to rest.
Let's start with step 1:
Step 1: Calculating the time taken to accelerate from rest to the maximum velocity
Given:
- Acceleration (a) = 3 m/s^2
- Initial velocity (u) = 0 (assuming the crane starts from rest)
- Final velocity (v) = ?
Using the formula:
v = u + at
Substituting the values:
v = 0 + (3 m/s^2) = 3 m/s
To calculate the time taken (t) to reach the maximum velocity, we can rearrange the formula:
t = (v - u) / a
Substituting the values:
t = (3 m/s - 0) / 3 m/s^2 = 1 second
So it takes 1 second to accelerate to the maximum velocity.
Step 2: Calculating the time taken to travel at a constant velocity
Since the crane lifts the units at a constant velocity past the second to fourth stories, there is no acceleration or deceleration during this stage. Each story is 3.5 m high, and there are 3 stories (second to fourth). So the total distance to be traveled at a constant velocity is:
distance = 3.5 m/story * 3 stories = 10.5 m
To calculate the time taken (t) to travel this distance at constant velocity, we can use the formula:
t = distance / v
Substituting the values:
t = 10.5 m / 3 m/s = 3.5 seconds
So it takes 3.5 seconds to travel at constant velocity through the second to fourth stories.
Step 3: Calculating the time taken to decelerate from the maximum velocity to rest
Again, we use the formula:
t = (v - u) / a
Substituting the values:
t = (0 - 3 m/s) / -3 m/s^2 = 1 second
So it takes 1 second to decelerate from the maximum velocity to rest.
Finally, we add up the times from each stage to find the total time taken:
Total time = time for acceleration + time for constant velocity + time for deceleration
Total time = 1 second + 3.5 seconds + 1 second
Total time = 5.5 seconds
Therefore, it takes 5.5 seconds for an air conditioning unit to reach the top of the fifth story.