Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1500 m due east and then turns due north and travels another 1430 m before reaching the campground. The second cyclist starts out by heading due north for 1870 m and then turns and heads directly toward the campground.
(a) At the turning point, how far is the second cyclist from the campground?
(b) What direction (measured relative to due east) must the second cyclist head during the last part of the trip?
To solve this problem, we can use the Pythagorean theorem and trigonometric functions. Let's break down the problem step by step.
(a) At the turning point, how far is the second cyclist from the campground?
For the first cyclist, we have a right-angled triangle with sides 1500 m and 1430 m. We can use the Pythagorean theorem to find the length of the hypotenuse (distance to the campground):
c1 = sqrt((1500^2) + (1430^2))
c1 ≈ 2042.46 m
For the second cyclist, we have a right-angled triangle with one leg of 1870 m. We need to find the length of the other leg to determine the distance to the campground. Since the two cyclists are heading toward the campground, we know the distance between them is the same at the turning point. Therefore, the length of the other leg is also 1430 m. Let's call the distance to the campground for the second cyclist c2:
c2 = sqrt((1870^2) + (1430^2))
c2 ≈ 2324.72 m
So at the turning point, the second cyclist is approximately 2324.72 m away from the campground.
(b) What direction (measured relative to due east) must the second cyclist head during the last part of the trip?
To find the direction, we can use trigonometric functions. We know the lengths of the two legs of the right-angled triangle formed by the second cyclist's path. Let's find the angle between the direction of due east and the second cyclist's path.
tan(theta) = (opposite/adjacent)
tan(theta) = (1430/1870)
Now, we can take the inverse tangent (arctan) of both sides to find the angle theta:
theta ≈ arctan(1430/1870)
theta ≈ 36.44 degrees
Therefore, the second cyclist must head approximately 36.44 degrees north of due east during the last part of the trip.
In summary:
(a) At the turning point, the second cyclist is approximately 2324.72 m away from the campground.
(b) The second cyclist must head approximately 36.44 degrees north of due east during the last part of the trip.
I will be glad to critique your work. We are getting more physics questions than we can handle in a timely manner, and have to be selective about who receives help.
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Your information on the first cyclist establishes the vector distance of campground from the starting point. Call that vector A. The distance traveled by the second biker on the first leg is vector B.
The answer to your both questions requires calculating the vector difference
A - B.