A circus performer begins his act by walking out along a nearly horizontal high wire. He slips and falls to the safety net, 25.0 ft below. The magnitude of his displacement from the beginning of the walk to the net is 26.7 ft. (a) How far out along the high wire did
he walk?
Two bicyclists, starting at the same place are riding toward the same campground by two different routes. One cyclist rides 1080 m due east and then turns due north and travels another 1430 m before reaching the campground.
To find the distance the circus performer walked along the high wire, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
Let's assume that the distance the performer walked along the high wire is "x" ft.
According to the problem, the magnitude of the displacement from the beginning of the walk to the net is 26.7 ft. This means the displacement is along the vertical direction (downward) and can be represented as the height of a right triangle formed by the hypotenuse (the displacement), the horizontal distance (x), and the vertical distance (25.0 ft).
Using the Pythagorean theorem, we can write the equation:
x^2 + 25.0^2 = 26.7^2
Simplifying the equation:
x^2 + 625 = 712.89
Subtracting 625 from both sides:
x^2 = 87.89
Taking the square root of both sides:
x ≈ √87.89
x ≈ 9.37 ft
Therefore, the circus performer walked approximately 9.37 ft out along the high wire.
To determine how far out along the high wire the performer walked, we need to find the horizontal distance between the starting point and the location where he fell. This horizontal distance is also known as the displacement.
Given that the magnitude of his displacement is 26.7 ft, we know that the displacement can be positive or negative, representing the direction of the walk. However, since he falls downwards to the safety net, we can assume that the displacement is negative.
Therefore, we have:
Displacement = -26.7 ft
Now, let's find the vertical distance between the starting point and the safety net. We are given that the performer falls 25.0 ft below the starting point. In this case, we can assume that the vertical distance is positive, considering the downward direction of the fall.
Vertical distance = 25.0 ft
Since the high wire is nearly horizontal, we can use the Pythagorean theorem to find the total distance (d) between the starting point and the location where he fell. The Pythagorean theorem states that the square of the hypotenuse (d) is equal to the sum of the squares of the other two sides.
So, according to the Pythagorean theorem:
d² = (horizontal distance)² + (vertical distance)²
d² = (displacement)² + (vertical distance)²
d² = (-26.7 ft)² + (25.0 ft)²
d² = 712.89 ft² + 625.0 ft²
d² = 1337.89 ft²
To find the value of d, we take the square root of both sides:
d = √(1337.89 ft²)
d ≈ 36.6 ft
Therefore, the performer walked approximately 36.6 ft out along the high wire before slipping and falling to the safety net.
x^2 + 25^2 = 26.7^2
x = 9.37 ft