When a soccer ball is kicked toward a player and the player...

When a soccer ball is kicked toward a player and the player deflects the ball by "heading" it, the acceleration of the head during the collision can be significant. Figure $2-38$ gives the measured acccleration $a(t)$ of a soccer player's head for a bare head and a helmeted head, starting from rest. The scaling on the vertical axis is set by $a_{s}=200 \mathrm{~m} / \mathrm{s}^{2}$. At time $t=7,0 \mathrm{~ms}$, what is the difference in the speed acquired by the bare head and the speed acquired by the helmeted head?

When a soccer ball is kicked toward a player and the player deflects the ball by "heading" it, the acceleration of the head
during the collision can be significant. Figure $2-38$ gives the measured acccleration $a(t)$ of a soccer player's head for a bare head and a helmeted head, starting from rest. The scaling on the vertical axis is set by $a_{s}=200 \mathrm{~m} / \mathrm{s}^{2}$. At time $t=7,0 \mathrm{~ms}$, what is the difference in the speed acquired by the bare head and the speed acquired by the helmeted head?

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Key Concepts

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It involves understanding relationships among acceleration, velocity, and displacement, where a key principle is that integrating acceleration over time gives the change in velocity.

Integration of Acceleration

Integration of the acceleration function over a given time interval provides the change in velocity of the object. Since acceleration is the derivative of velocity, the area under an acceleration versus time graph represents the total change in speed, making it essential for solving problems in which the acceleration varies with time.

Graphical Analysis

Graphical analysis requires interpreting data from graphs, such as an acceleration versus time plot. By examining the area under the curve and applying appropriate scaling factors, one can quantitatively determine changes in velocity, which is critical for comparing different scenarios or samples in dynamic events.

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