To determine when the two elevators cross each other and the distance each travels, we can use the principles of kinematics. Since both elevators start from rest, we can apply the equations of motion to find their positions over time. Let's break this down step by step.
Understanding the Problem
We have two elevators:
- The first elevator is moving upwards with an acceleration of 0.3 m/s².
- The second elevator is moving downwards with an acceleration of 0.15 m/s².
They start 60 meters apart, and we need to find out when they meet and how far each has traveled by that time.
Setting Up the Equations
We can use the following kinematic equation to describe the position of each elevator:
s = ut + (1/2)at²
Where:
- s = distance traveled
- u = initial velocity (which is 0 for both elevators)
- a = acceleration
- t = time
Position of the Upward Moving Elevator
For the upward-moving elevator:
- Initial velocity (u) = 0
- Acceleration (a) = 0.3 m/s²
Using the equation:
s₁ = 0 + (1/2)(0.3)t² = 0.15t²
Position of the Downward Moving Elevator
For the downward-moving elevator:
- Initial velocity (u) = 0
- Acceleration (a) = 0.15 m/s²
Using the equation:
s₂ = 60 - (1/2)(0.15)t² = 60 - 0.075t²
Finding the Time of Intersection
To find when the elevators cross each other, we set their positions equal:
0.15t² = 60 - 0.075t²
Combining the terms gives:
0.15t² + 0.075t² = 60
0.225t² = 60
Now, solving for t²:
t² = 60 / 0.225
t² = 266.67
Taking the square root:
t ≈ 16.33 seconds
Calculating the Distance Traveled
Now that we have the time, we can find the distance each elevator has traveled:
Distance of the Upward Moving Elevator
Using the time in the first elevator's equation:
s₁ = 0.15(16.33)²
s₁ ≈ 0.15 × 266.67 ≈ 40 meters
Distance of the Downward Moving Elevator
Now for the downward-moving elevator:
s₂ = 60 - 0.075(16.33)²
s₂ = 60 - 0.075 × 266.67
s₂ ≈ 60 - 20 ≈ 40 meters
Summary of Results
In conclusion, the elevators cross each other after approximately 16.33 seconds, and each elevator travels about 40 meters during this time. This example illustrates how kinematic equations can be applied to analyze motion in a straightforward manner.
