O Instructions The module shows four graphs: the first plots a moving object's position at time t; the second shows the object's velocity (the derivative of the position function); the third show the object's acceleration (the derivative of the velocity function, or the second derivative of the position function); and the fourth shows the total distance traveled. The position function is graphed in black when the position of the object is positive and red when it is negative. To the left of the position graph, a moving dot demonstrates the motion represented by the position function. To the right, vertical bars keep track of the distance traveled, with orange for motion in the positive direction and purple for motion in the negative direction. Click the Animate button to observe the different graphs being drawn simultaneously as the object moves. After the graphs are drawn, moving the mouse over any graph traces a vertical line in each diagram so that you may compare key points of the graphs at a given time. You can select either a polynomial or a trigonometric position function form the pull-down menu at the top. Drag the slider handles, click on a slider bar or click on number next to a slider to adjust the constants in the equations. The resulting equation is shown at the upper right and the graphs will update accordingly. O Simulation THE DYNAMICS OF LINEAR MOTION 1 flx) 33- Position CONCEPT Polynomial YOU WILL LEARN ABOUT: Linear motion as shown by position, velocity, and acceleration functions. Animate -31- 1 y = a, + a¡t+ a,t² + azt³ + a¸t+ 2 3 5 7 ↑ f'(x). Velocity = 0 +41 – 4.5² + 0.56³ + 0A 44 ao -20 -10 10 20 -15- 3 5 8 aj 4 f"(x). Acceleration 22- -10 -5 10 a2 -4.5 Zurijeta/Shutterstock.com -5 -2.5 2.5 5 The position of a ball thrown straight up in the air can be modeled with a polynomial -13- az 0.56 1 2. 3 4 5 8 -1 -0.5 0.5 a4 function. If s = f (1) is the position -1 -0.5 0.5 Total Distance As function of the ball, then At 85- represents the average velocity WHEN WOULD I USE THIS 3 5 7 INSTRUCTIONS EXERCISES Click here to access the simulation in a new window. O Exercise The harmonic motion of a particle is given by f(t) = 2 cos(3t) + 3 sin(2t), 0 s ts 8. (a) When is the position function decreasing? (Round your answers to one decimal place. Enter your answer using interval notation.) (b) During how many time intervals is the particle's acceleration positive? time intervals (c) At what time is the particle at the farthest distance away from its starting position in the negative direction? (Round your answer to one decimal place.) t = How far away is it from its original position? (Round your answer to the nearest integer.)