how to solve this? the answer key says 10 floors (11 + the ground floor)
An elevator starts from rest at the ground floor of the building. It experiences an acceleration a=0.500m/s^2 and reaches a velocity of 5.48m/s. What is the floor reached by the moving elevator, knowing that each floor us 3.00m high?
To find the floor reached by the moving elevator, we can use the following steps:
Step 1: Determine the time taken for the elevator to reach its final velocity.
- We can use the equation v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
- Plugging in the given values, we have 5.48 m/s = 0 + 0.500 m/s^2 * t.
- Solving for t, we have t = 5.48 m/s / 0.500 m/s^2 = 10.96 seconds.
Step 2: Find the distance traveled by the moving elevator.
- We can use the equation s = ut + (1/2)at^2, where s is the distance traveled.
- Plugging in the given values, u = 0 m/s (initial velocity), a = 0.500 m/s^2 (acceleration), and t = 10.96 seconds (time taken), we have:
s = 0 * 10.96 + (1/2) * 0.500 * (10.96)^2.
- Simplifying this, we get s = 0 + (0.5) * 0.500 * 120.1216.
- Evaluating further, we have s = 30.03 meters (distance traveled).
Step 3: Calculate the floor reached by the moving elevator.
- Since each floor is 3.00 meters high, we can divide the distance traveled by the height of each floor to find the number of floors.
- Taking 30.03 meters / 3.00 meters per floor, we get 10.01 floors.
- Rounding up to the nearest whole number, we have 10.01 ≈ 10 floors.
Therefore, the floor reached by the moving elevator is 10 floors (including the ground floor).
To solve this problem, we need to use the equations of motion to find the time taken by the elevator to reach its final velocity.
First, let's find the time taken by the elevator to reach a velocity of 5.48 m/s. We can use the following equation:
v = u + at
where v is the final velocity, u is the initial velocity (which is 0 m/s as the elevator starts from rest), a is the acceleration, and t is the time taken.
Plugging in the given values:
5.48 m/s = 0 m/s + (0.500 m/s^2)t
Now we can solve for t:
t = (5.48 m/s - 0 m/s) / 0.500 m/s^2
t = 10.96 s
So, it takes the elevator 10.96 seconds to reach a velocity of 5.48 m/s.
Next, to find the distance traveled by the elevator, we can use the equation:
s = ut + (1/2)at^2
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
Plugging in the values:
s = 0 m/s * 10.96 s + (1/2) * 0.500 m/s^2 * (10.96 s)^2
Simplifying this equation gives us:
s = (1/2) * 0.500 m/s^2 * (120.1216 s^2)
s = 30.0304 m
Now, since each floor is 3.00 m high, we can divide the distance traveled by the elevator by the height of each floor to find the floor reached.
Floor reached = s / height of each floor
Floor reached = 30.0304 m / 3.00 m
Floor reached = 10.01
So, the elevator reaches the 10th floor (11th floor including the ground floor).