Three leaves are falling from a tree. Figure 1 shows the way a leaf falls. Figure 2 illustrates the starting orientation of the three leaves. If the blue colour leaf is at 200 cm, the red colour leaf is at 425 cm and the yellow colour leaf is at 375 cm from the ground, then what would be the orientation of all the leaves when they reach the ground?
Show Hint
For problems involving repeating cycles, first clearly define the rule of the cycle. Then, for each case, determine how many full cycles and partial cycles occur. Here, the number of 50 cm intervals determines the number of flips.
Step 1: Understanding the Concept:
The problem requires us to apply a repeating pattern of motion to three different objects, each starting from a different initial condition. We need to determine the final state of each object after it has fallen a specific distance.
Step 2: Key Formula or Approach:
From Figure 1, we can establish the rule for the leaf's orientation change:
The leaf starts in an initial orientation.
After falling 50 cm, its orientation is flipped horizontally.
After falling another 50 cm (100 cm total), it flips back to its original orientation.
This cycle repeats. So, the orientation flips every 50 cm.
If a leaf falls a distance \(d\), the number of flips it undergoes is \(N = \lfloor d / 50 \rfloor\).
If \(N\) is even, the final orientation is the same as the original.
If \(N\) is odd, the final orientation is the horizontally flipped version of the original.
Step 3: Detailed Explanation:
Let's calculate the number of flips for each leaf. The distance fallen is equal to its starting height. Blue Leaf:
Starting height = 200 cm.
Distance fallen, \(d\) = 200 cm.
Number of flips, \(N = \lfloor 200 / 50 \rfloor = 4\).
Since 4 is an even number, the blue leaf will end in its original orientation.
Red Leaf:
Starting height = 425 cm.
Distance fallen, \(d\) = 425 cm.
Number of flips, \(N = \lfloor 425 / 50 \rfloor = \lfloor 8.5 \rfloor = 8\).
Since 8 is an even number, the red leaf will end in its original orientation.
Yellow Leaf:
Starting height = 375 cm.
Distance fallen, \(d\) = 375 cm.
Number of flips, \(N = \lfloor 375 / 50 \rfloor = \lfloor 7.5 \rfloor = 7\).
Since 7 is an odd number, the yellow leaf will end in its flipped orientation.
4. Matching with Options:
Let's check the options against our findings:
Blue: Original orientation (pointing down-left).
Red: Original orientation (pointing up-right).
Yellow: Flipped orientation (original was down-right, so flipped is down-left).
Option (C) is the only one that shows the blue leaf and red leaf in their original orientations, and the yellow leaf in its flipped orientation.
Step 4: Final Answer:
Based on the analysis, the correct final arrangement of the leaves is shown in option (C).
