The center of mass G of a high jumper follows the trajectory...

The center of mass $G$ of a high jumper follows the trajectory shown. Determine the component $v_{0}$ measured in the vertical plane of the figure, of his takeoff velocity and angle $\theta$ if the apex of the trajectory just clears the bar at $A$. (In general, must the mass center $G$ of the jumper clear the bar during a successful jump?)

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Key Concepts

Center of Mass Dynamics

The center of mass (COM) is the point where the mass of the body is considered to be concentrated and follows the projectile path. In high jump, while the COM typically follows this parabolic trajectory, the jumper can manipulate their body orientation to clear the bar even if the COM does not pass over it. Understanding the dynamics of the COM is important in explaining how body movements contribute to successful high jumps.

Decomposition of Velocity

Decomposing the initial takeoff velocity into vertical and horizontal components is essential for understanding projectile motion. The vertical component is responsible for reaching the maximum height (which is critical in high jumping), and the horizontal component determines the range of the jump. The angle of takeoff relates these two components and is used to compute the required initial speed to achieve the necessary trajectory.

Projectile Motion

Projectile motion refers to the motion of an object under the influence of gravity alone after an initial thrust. It involves the use of kinematic equations to determine the trajectory, where the vertical motion is influenced by gravity while the horizontal component remains constant. This concept is crucial for analyzing the jumper's arc, particularly the apex that just clears the bar.

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