University of Babylon College of Engineering Mechanical Engineering Dept. Subject : Mathematics III Class : 2 nd First Semester Year :

Transcription

1 Uniersity of Babylon College of Engineering Mechanical Engineering Dept. Subject : Mathematics III Class : nd First Semester Year : 16-17

2 VECTOR FUNCTIONS

3 SECTION 13. Ideal Projectile Motion

4 Ideal Projectile Motion What do we mean by the word ideal here? Assumptions: The projectile behaes like a particle moing in a ertical coordinate plane. The only force acting on the projectile during its flight is the constant force of graity. The projectile will follow a trajectory that is perfectly parabolic.

5 Ideal Projectile Motion Position, elocity, acceleration due to graity, and launch angle at t = Initial elocity: sin j cos i sin j cos i sin j Initial position: r i j r = at time t = cosi a gj

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7 o Newton s second law of motion: f f ma dr m dt The force of graity is: f mg Force is in the downward direction j

8 Newton s second law of motion: the force acting on the projectile is equal to the projectile s mass times its acceleration. For ideal motion, this force is solely the graitational force: m d dt r mgj d dt r gj a gj

9 Ideal Projectile Motion Position, elocity, and acceleration at a later time t xy, r xi yj a gj R Horizontal range

10 m Ideal Projectile Motion Newton s second law of motion: the force acting on the projectile is equal to the projectile s mass times its acceleration. For ideal motion, this force is solely the graitational force: d dt r mgj Let s sole this initial alue problem, using initial conditions: and First integration: Second integration: d dt r gj dr r r dt when t dr gt j dt 1 r j r gt t

11 Preious equations: Ideal Projectile Motion 1 r j r gt t cos i sin j Substitution: r i j 1 r j cos i sin j gt t t 1 cos ti sin t gt j

12 Ideal Projectile Motion t t 1 gt r cos i sin j This is the ector equation for ideal projectile motion. The angle is the projectile s launch angle (firing angle, angle of eleation), and is the projectile s initial speed. Break into a pair of scalar equations: y sin t gt x cos t These are the parametric equations for ideal projectile motion. Graity constants: m ft g sec sec 1

13 Ideal Projectile Motion t t 1 gt r cos i sin j y sin t gt x cos t If the ideal projectile is fired from the point of the origin: cos 1 x x t y y sin t gt x, y 1 instead

14 Practice Problem A projectile is fired from the origin oer horizontal ground at an initial speed of 5 m/sec and a launch angle of 6 degrees. Where will the projectile be 1 sec later? Parametric equations for this situation, ealuated at this time: x 5cos y 5sin Ten seconds after firing, the projectile is about m in the air and 5 m downrange

15 Height Flight Time Range

16 Height, Flight Time, and Range The projectile reaches its highest point when its ertical elocity component is zero: dy dt 1 y sin t gt sin gt t sin g Height at this time: sin 1 sin y sin g g g sin g g sin g sin g

17 Height, Flight Time, and Range To find the total flight time of the projectile, find the time it takes for the ertical position to equal zero: 1 y sin t gt 1 t sin gt sin t, t g

18 Height, Flight Time, and Range To find the projectile s range R, the distance from the origin to the point of impact on horizontal ground, find the alue of x when t equals the flight time: R cos t g sin sin g x cos t sincos g sin g sin 1 45 Note: The range is largest when or.

19 Height, Flight Time, and Range For ideal projectile motion when an object is launched from the origin oer a horizontal surface with initial speed and launch angle : Maximum Height: Flight Time: Range: R t y max sin g sin g sin g

20 Practice Problems Find the maximum height, flight time, and range of a projectile fired from the origin oer horizontal ground at an initial speed of 5 m/sec and a launch angle of 6 degrees. Then graph the path of the projectile. 5sin 6 Max Height: y max 9.8 m 5 sin 6 Flight Time: t sec 5 Range: R sin1, m

21 Practice Problems Find the maximum height, flight time, and range of a projectile fired from the origin oer horizontal ground at an initial speed of 5 m/sec and a launch angle of 6 degrees. Then graph the path of the projectile. Parametric equations for the path of the projectile: x 5cos6 t 5t 1 y 5sin 6 t 9.8 t 5 3 t4.9t Graph window: [, 5] by [, 1]

22 y Practice Problems To open the 199 Summer Olympics in Barcelona, bronze medalist archer Antonio Rebollo lit the Olympic torch with a flaming arrow. Suppose that Rebollo shot the arrow at a height of 6 ft aboe ground leel 3 yd from the 7-ft-high cauldron, and he wanted the arrow to reach a maximum height exactly 4 ft aboe the center of the cauldron. 6 y max 74 x 9

23 Practice Problems y (a) Express max in terms of the initial speed and firing angle. y max (b) Use alue of. sin g ymax 74 sin ft and the result from part (a) to find the 6 74 sin 6 3 sin sin 435

24 Practice Problems cos x x cos t (c) Find the alue of. Values when the arrow reaches maximum height: 9 cos sin g cos 3 cos

25 Practice Problems (d) Find the initial firing angle of the arrow. sin 435 Put these parts together: 88 cos 435 sin cos tan tan