Graphical Integration in Motion Analysis In a forward punch...

Graphical Integration in Motion Analysis 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 $v_{s}=8.0 \mathrm{m} / \mathrm{s}$ . How far has the fist moved at (a) time $t=50 \mathrm{ms}$ and (b) when the speed of the fist is maximum?

Graphical Integration in Motion Analysis
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 $v_{s}=8.0 \mathrm{m} / \mathrm{s}$ . How far has the fist moved at (a) time $t=50 \mathrm{ms}$ and (b) when the speed of the fist is maximum?

Key Concepts

Displacement Calculation via Integration

Displacement in motion analysis is computed by integrating velocity with respect to time. This process accumulates the small contributions of velocity over time to yield the total change in position. The method highlights the fundamental relationship between a function and its integral, where the area under the velocity curve directly corresponds to the distance traveled.

Graphical Integration

Graphical integration involves calculating the area under a curve, which in the context of a velocity-time graph, represents the displacement of an object. This method is useful for estimating displacement when an analytical integration may be difficult or when data is available only in graphical form.

Velocity-Time Graph

A velocity-time graph is a graphical representation where velocity is plotted on the vertical axis and time on the horizontal axis. It is used to analyze motion; by examining the shape and area under the curve, one can determine important characteristics such as changes in speed and acceleration, which are directly related to the object's motion.

Transcript

00:01 So here, the key thing to remember here is that the position or the total displacement is simply the area beneath the curve of a velocity versus time graph.

00:16 So we can say that essentially the position is simply the integral of velocity.

00:22 Again, the integral being simply the area under the curve.

00:26 So we can say x sub 1 minus x not you can also say delta x however you would like to represent it this would simply be equal to the area between the velocity curve and the time axis from t sub 1 or other t sum t not or t initial to t sub 1 or again delta t so we can say that to compute the position and of the fist at t equals 50 milliseconds we just divide the area so for part a we can say that the area of part a would be simply equal to a triangle so one half times 0 .010 seconds multiplied by two meters per second this is equaling 0 .01 meter now for we can say the area sub b would be equal to here it would be a trapezoid so we can say 1 1 1�t times 0 .040 seconds.

01:49 This would be 10 milliseconds to 50 milliseconds.

01:53 So essentially 50 milliseconds minus 10 milliseconds.

01:57 This is equaling 0 .4 .040 seconds.

02:01 And then multiplied by 2 plus 4 units here of course meters per second.

02:05 This is equaling 0 .12 meters.

02:09 And then substituting these values into equation 230, we can have x sub 1 minus 0, equaling 0 plus 0 .01 plus 0 .12.

02:24 This is equaling 0 .13 meters or this is simply equaling x sub 1.

02:32 This would be our final answer for part a...