On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
Question
On an open ground, a motorist follows a track that turns to his left by an angle of 60° after every 500 m. Starting from a given turn, specify the displacement of the motorist at the third, sixth and eighth turn. Compare the magnitude of the displacement with the total path length covered by the motorist in each case.
Solution
Sure, let's solve this step by step.
- Displacement after the third turn:
The motorist makes a turn every 500m, so after three turns, he has traveled 3*500 = 1500m. However, because he turns 60° to the left at each turn, the direction of his travel changes. After three turns, he has turned a total of 180°, which means he is now traveling in the opposite direction from where he started. Therefore, his displacement (which is a vector quantity that takes into account both distance traveled and direction) is 0.
- Displacement after the sixth turn:
After six turns, the motorist has traveled 6*500 = 3000m. He has also made six 60° turns, for a total of 360°. This means he has made a full circle and is now traveling in the same direction as when he started. Therefore, his displacement is again 0.
- Displacement after the eighth turn:
After eight turns, the motorist has traveled 8500 = 4000m. He has also made eight 60° turns, for a total of 480°. This is equivalent to a full circle (360°) plus an additional 120°. Therefore, he is traveling in a direction that is 120° to the left of his starting direction. To find his displacement, we can use the law of cosines. The displacement will be √[(500^2) + (500^2) - 2500500cos(120°)] = 866.03m approximately.
Comparing the magnitude of the displacement with the total path length:
- After the third turn, the displacement is 0 while the total path length is 1500m.
- After the sixth turn, the displacement is 0 while the total path length is 3000m.
- After the eighth turn, the displacement is approximately 866.03m while the total path length is 4000m.
In each case, the displacement is less than the total path length because the motorist is changing direction as he travels. The difference between displacement and total path length is most pronounced after the third and sixth turns, when the motorist has traveled a significant distance but his change in direction means that his starting and ending points are the same, resulting in a displacement of 0.
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