An elevator in a high rise building accelerates and decelerates at the rate of 1 foot per second squared. Its maximum speed is 8 feet per second. It starts from rest, accelerates to its maximum speed and stays at that maximum speed until it approaches its destination where it decelerates to rest.
How far does the elevator travel if it accelerates from rest until it reaches its maximum speed?
How far has the elevator traveled after starting from rest and traveling for 10 seconds without decelerating?
not calculus
it's for calculus
To calculate the distance traveled by the elevator when it accelerates from rest until it reaches its maximum speed, we can use the formula for distance traveled under constant acceleration:
Distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)
Given:
Initial velocity (u) = 0 feet per second (since it starts from rest)
Acceleration (a) = 1 foot per second squared
Since the elevator is accelerating from rest, the initial velocity is 0. We also know that the maximum speed is 8 feet per second.
To calculate the time it takes for the elevator to reach its maximum speed, we can use the formula:
Time = (Final velocity - Initial velocity)/Acceleration
Given:
Final velocity (v) = 8 feet per second
Acceleration (a) = 1 foot per second squared
Time = (8 - 0) / 1 = 8 seconds
Now we can substitute the values into the distance formula:
Distance = (0 * 8) + (0.5 * 1 * 8^2)
Distance = 0 + (0.5 * 1 * 64)
Distance = 0 + (0.5 * 64)
Distance = 0 + 32
Distance = 32 feet
Therefore, the elevator travels 32 feet when it accelerates from rest until it reaches its maximum speed.
To determine how far the elevator has traveled after starting from rest and traveling for 10 seconds without decelerating, we need to consider two separate scenarios:
1. The elevator reaches its maximum speed within 10 seconds:
If the elevator reaches its maximum speed within 10 seconds, it will remain at that speed until it decelerates to rest. In this case, the distance traveled can be calculated using the formula:
Distance = (Initial velocity * Time) + (0.5 * Acceleration * Time^2)
Given:
Initial velocity (u) = 0 feet per second (since it starts from rest)
Acceleration (a) = 1 foot per second squared
Time (t) = 10 seconds
Distance = (0 * 10) + (0.5 * 1 * 10^2)
Distance = 0 + (0.5 * 1 * 100)
Distance = 0 + 50
Distance = 50 feet
Therefore, if the elevator reaches its maximum speed within 10 seconds, it will have traveled a distance of 50 feet.
2. The elevator does not reach its maximum speed within 10 seconds:
If the elevator does not reach its maximum speed within 10 seconds, it will still be in the process of accelerating. In this case, we need to calculate the distance traveled during the acceleration phase and add it to the distance traveled at the maximum speed.
Distance during acceleration = (0 * t) + (0.5 * a * t^2)
Distance during acceleration = 0 + (0.5 * 1 * 10^2)
Distance during acceleration = 0 + 50
Distance during acceleration = 50 feet
Since the elevator reaches its maximum speed after 8 seconds (as calculated earlier), we can calculate the distance traveled at the maximum speed using the formula:
Distance at maximum speed = Speed * Time
Distance at maximum speed = 8 * (10 - 8)
Distance at maximum speed = 8 * 2
Distance at maximum speed = 16 feet
Now we can add the distance traveled during acceleration to the distance traveled at the maximum speed:
Total distance = Distance during acceleration + Distance at maximum speed
Total distance = 50 + 16
Total distance = 66 feet
Therefore, if the elevator does not reach its maximum speed within 10 seconds, it will have traveled a distance of 66 feet.