Graphical Integration in Motion Analysis Figure 2-15 a gives...

Graphical Integration in Motion Analysis
Figure $2-15 a$ gives the acceleration of a volunteer's head and torso during a rear-end collision. At maximum head acceleration, what is the speed of (a) the head and (b) the torso?

Key Concepts

Graphical Integration

Graphical integration is a technique used to determine the net change of a quantity by finding the area under its rate-of-change graph. In motion analysis, this method is particularly useful for converting acceleration versus time graphs into velocity information by measuring the area under the curve within specified time intervals.

Acceleration

Acceleration is the rate at which an object’s velocity changes with time. It is a fundamental concept in kinematics and is essential for understanding how forces affect motion. In problems involving motion analysis, acceleration data is often integrated to derive velocity and displacement.

Velocity

Velocity is the speed of an object in a specific direction, representing the first derivative of displacement with respect to time. In motion analysis, obtaining velocity from acceleration involves integrating the acceleration-time graph, which helps in predicting the object’s state of motion at various instances.

Kinematics

Kinematics is the branch of mechanics that describes the motion of objects without considering the forces that cause this motion. It involves parameters such as displacement, velocity, acceleration, and time, providing the mathematical framework needed to analyze and predict the behavior of moving bodies.

Transcript

00:01 So here we can say that the velocity here, velocity sub 1 minus the initial velocity, the velocity at any point in the graph minus the initial velocity, so the change in the velocity since the beginning.

00:17 This would be equaling to the area under the curve of an acceleration versus time graph.

00:34 And this is of course because the velocity is going to be the integral of the acceleration.

00:40 So for part a, we can see from figure 215a, the head begins to accelerate from rest.

00:48 And this would be velocity, the initial velocity of course, zero.

00:55 Time is equaling 110 milliseconds.

00:59 And this reaches a maximum value of 90 meters per se.

01:06 Squared at t sub 1 equaling 160 milliseconds.

01:16 So we can say that the area of this region would be equaling one half multiplied by 160, my apologies, 160 minus 110 times 10 to the negative third seconds multiplied by 90 meters per second squared and this is giving us 2 .25 meters per second and this is of course the value v sub 1 2 .25 meters per second at t sub 1 so this would be our answer for part a and for part b we know that to compute the speed of the torso at t sub 1 equaling 160 milliseconds we can divide the area into four regions so we can say from 0 seconds to 40 milliseconds, we can say region a has no area.

02:34 And then we can say from 40 milliseconds to 100 milliseconds, we can say that region b, the area of region b would be similar to the area of a triangle or one -half times the base times height.

02:49 So 1 .5 times 0 .06000 seconds multiplied by 50 .0 meters per second squared.

02:59 And this is giving us 1 .50 meters per second.

03:04 And then for area c, this would be 100 to 120 milliseconds.

03:11 And this has the shape of a rectangle.

03:14 So this would be 0 .0200 seconds...

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