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The grand jeté is a classic ballet maneuver in which a dancer executes a horizontal leap while moving her arms and legs up and then down. At the center of the leap, the arms and legs are gracefully extended, as we see in Figure \(\mathrm{P} 7.81 \mathrm{a}\). The goal of the leap is to create the illusion of flight. As the dancer moves through the air, he or she is in free fall. In Chapter \(3,\) we saw that this leads to projectile motion. But what part of the dancer follows the usual parabolic path? It won't come as a surprise to learn that it's the center of gravity. But when you watch a dancer leap through the air, you don't watch her center of gravity, you watch her head. If the translational motion of her head is horizontal-not parabolicthis creates the illusion that she is flying through the air, held up by unseen forces. Figure \(\mathrm{P} 7.81 \mathrm{b}\) illustrates how the dancer creates this illusion. While in the air, she changes the position of her center of gravity relative to her body by moving her arms and legs up, then down. Her center of gravity moves in a parabolic path, but her head moves in a straight line. It's not flight, but it will appear that way, at least for a moment. To perform this maneuver, the dancer relies on the fact that the position of her center of gravity A. Is near the center of the torso. B. Is determined by the positions of her arms and legs. C. Moves in a horizontal path. D. Is outside of her body.

Step by step solution

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Option Analysis

Let's analyze each option:A. Is near the center of the torso: It is true that normally the center of gravity of a person is near the center of the torso, however, this does not help to create the illusion of flight in the ballet maneuver.B. Is determined by the positions of her arms and legs: This statement is true and it is relevant to the ballet maneuver. By moving her arms and legs up and down, the dancer changes the position of her center of gravity, which helps to create the illusion of flight.C. Moves in a horizontal path: If the center of gravity moved in a horizontal path, the dancer wouldn't be able to perform a leap in the air, which contradicts the statement given.D. Is outside of her body: It is not possible for the center of gravity to be outside of her body.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion

Understanding the concept of projectile motion is crucial when analyzing movements like the grand jeté in ballet. Imagine throwing a ball at an angle into the air; its path represents projectile motion. It's a combination of two separate motions: horizontal velocity and vertical velocity affected by gravity. These motions are independent of each other.

In the case of a dancer performing a grand jeté, the dancer propels themselves off the ground, giving them horizontal velocity while gravity pulls them down, adding vertical velocity. This results in a curved path through the air. The principles of projectile motion help us predict where the dancer will land and the trajectory they will follow. Moreover, tweaking the legs and arms' position can subtly alter the center of gravity, affecting the motion's arc.

Free Fall

When we speak of free fall, we're talking about an object moving under the influence of gravity alone. It's a state of motion that occurs when the only force acting upon an object is gravity, resulting in a constant downward acceleration. An important aspect of free fall is that all objects accelerate downwards at the same rate, regardless of their mass, when air resistance is negligible.

In the grand jeté, the dancer is in free fall once they've left the ground. No other forces are acting on their body except for gravity. This is why, regardless of the dancer's size or mass, the downward acceleration they experience due to gravity is the same – roughly 9.8 m/s² towards the Earth. Understanding this helps us appreciate why the dancer's center of gravity follows a parabolic trajectory, even as they strive to maintain a horizontal gaze to strengthen the illusion of flight.

Parabolic Trajectory

The term parabolic trajectory is used to describe the path that an object follows when it's launched into the air and is subject to gravity's pull, assuming there's no air resistance. This parabolic shape is a consequence of the constant acceleration due to gravity and the initial horizontal velocity imparted to the object.

In ballet, the dancer's center of gravity describes a parabolic arc during a grand jeté. Even though their muscles exert force to leap, the center of gravity's path is determined by gravitational pull and their initial jump velocity, resulting in this characteristic 'curved' motion. By carefully coordinating body movements, the ballet dancer can manipulate their limbs to change the body's moment of inertia, thus making the motion of their center of gravity different from that of the head, creating a captivating performance that seems to defy gravity.