In a forward punch in karate, the fist begins at rest at the...

In a forward punch in karate, the fist begins at rest at the waist and is brought rapidly forward until the arm is fully extended. The speed $v(t)$ of the fist is given in Fig. $2-37$ for someone skilled in karate. The vertical scaling is set by $v_{x}=8.0 \mathrm{~m} / \mathrm{s}$. How far has the fist moved at (a) time $t=50 \mathrm{~ms}$ and (b) when the speed of the fist is maximum?

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Key Concepts

Integration in Motion Analysis

Integration is the mathematical process used to sum an infinite number of small contributions to determine a total quantity. In the context of motion, integrating the velocity function with respect to time gives the displacement, which is crucial when the velocity is not constant.

Graphical Data Interpretation

Interpreting graphs is a key skill in physics, as it involves reading and extracting quantitative information such as time intervals and maximum values from plotted curves. This skill enables the determination of specific details of a motion based on a visual representation.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. It involves relationships between displacement, velocity, and acceleration, and provides methods—like integration—to analyze motion over time.

Area Under a Velocity-Time Graph

The concept of finding the displacement by calculating the area under a velocity-time graph is central to kinematics. The area represents the integral of velocity over time, which yields the total distance traveled by an object during a certain time interval.

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