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Simulating Motion with Parametric Equations

Simulating Motion with Parametric Equations. Sec. 6.3b is getting interesting…. We’ll start right in with an example…. Morris the math mole digs along a horizontal line with the coordinate of his position (in feet) given by. Use parametric equations and a grapher to simulate his motion.

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Simulating Motion with Parametric Equations

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  1. Simulating Motion with Parametric Equations Sec. 6.3b is getting interesting…

  2. We’ll start right in with an example… Morris the math mole digs along a horizontal line with the coordinate of his position (in feet) given by Use parametric equations and a grapher to simulate his motion. Estimate the times when Morris changes direction. First, let’s look at: (an arbitrary number)

  3. We’ll start right in with an example… Morris the math mole digs along a horizontal line with the coordinate of his position (in feet) given by Use parametric equations and a grapher to simulate his motion. Estimate the times when Morris changes direction. Next, let’s look at:

  4. More Practice Problems… A rugby ball is kicked from a spot 2 feet above the ground straight up with an initial velocity of 46 ft/sec. Graph the ball’s height against time, find the height of the ball at 1, 2, and 3 sec, and calculate how long the ball is in the air. The general equation for vertical position of a projectile: Initial velocity Initial height

  5. More Practice Problems… A rugby ball is kicked from a spot 2 feet above the ground straight up with an initial velocity of 46 ft/sec. Graph the ball’s height against time, find the height of the ball at 1, 2, and 3 sec, and calculate how long the ball is in the air. Try graphing these: Huh? y(1) = 32 ft, y(2) = 30 ft, y(3) = – 4 ft The ball is in the air for approximately 2.918 sec.

  6. More Practice Problems… p.530-531: 38 – Capture the Flag (a) Check the graph  window [0,100] by [–1,10] Use simultaneous mode, and note that it’s the process of graphing that’s important, not the final graph… (b) Who capture the flag, and by how many feet? The faster runner captures the flag when the slower runner is still 4.1 feet away from the flag.

  7. More Practice Problems… p.531: 40 – Height of a Pop-up (a) Write an equation that models the height of the ball as a function of time t. (b) Use parametric mode to simulate the pop-up. Graph and trace: (c) Use parametric mode to graph the height against time. Graph and trace:

  8. More Practice Problems… p.531: 40 – Height of a Pop-up (d) How high is the ball after 4 sec? Solve both graphically, and algebraically: The ball is 69 ft above the ground after 4 sec. (e) What is the maximum height of the ball? How many seconds does it take to reach its maximum height? Solve graphically… When t = 2.5 sec, the ball is at it maximum height of 105 ft.

  9. Projectile Motion with Parametric Equations Our last new stuff in Sec. 6.3

  10. To this point, we’ve talked about motion in only one direction (i.e., along only one axis)… Now, imagine a baseball thrown from a point y feet above ground level with an initial speed of v ft/sec at an angle 0 with the horizontal: 0 0 y What is the component form of the initial velocity? v 0 v sin 0 0 0 y 0 v cos 0 v = v cos 0, v sin 0 0 0 0 0 x

  11. Now, imagine a baseball thrown from a point y feet above ground level with an initial speed of v ft/sec at an angle 0 with the horizontal: 0 0 y The horizontal and vertical components of an object’s motion in this situation are independent of each other, and can be modeled by the following parametric equations: v 0 v sin 0 0 0 y 0 v cos 0 0 x

  12. Our first example: Kevin hits a baseball at 3 ft above the ground with an initial speed of 150 ft/sec at an angle of 18 with the horizontal. Will the ball clear a 20-ft wall that is 400 ft away? Equations modeling the path of the ball: When with the ball reach the wall?  After 2.804 sec. What is the height of the ball at this point?  y = 7.178 ft. Can we use our calculators to solve this???

  13. More Practice… Proctor kicks for points in a rugby game. He place-kicks the ball with an initial speed of 63 ft/sec at an angle of 46 with the horizontal. If the ball heads directly towards the 10 ft-high crossbar that is 116 ft from Proctor, will the ball clear the crossbar? The equations: Time it takes the ball to reach the crossbar:

  14. More Practice… Proctor kicks for points in a rugby game. He place-kicks the ball with an initial speed of 63 ft/sec at an angle of 46 with the horizontal. If the ball heads directly towards the 10 ft-high crossbar that is 116 ft from Proctor, will the ball clear the crossbar? Height of the ball at this time: No, the ball will fall short by 2.290 ft. Can we solve graphically?

  15. More Practice… Starting with the same situation as the previous example, there is now a 6 ft/sec split-second wind gust just as Proctor kicks the ball. If the wind acts in the horizontal direction behind the ball, will the kick clear the crossbar? The new equations: Yes, the ball will clear the crossbar by 8.699 ft.

  16. More Practice… Tony and Sue are launching yard darts 20 ft from the front edge of a circular target of radius 18 in. on the ground. If Sue throws the dart directly at the target, and releases it 4 ft above the ground with an initial velocity of 25 ft/sec at a 55 angle, will the dart hit the target? The equations: The dart lands when y = 0… Find the time when this happens:

  17. More Practice… Tony and Sue are launching yard darts 20 ft from the front edge of a circular target of radius 18 in. on the ground. If Sue throws the dart directly at the target, and releases it 4 ft above the ground with an initial velocity of 25 ft/sec at a 55 angle, will the dart hit the target? Now, find how far the dart traveled in this time: So, will the dart land in the target? Yes, the dart lands about 10 inches past the front edge of the target  Refer to this example when you do #47 in the HW!!!

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