CHAPTER 4 Motion in 2D and 3D

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1 General Physics 1 (Phys : Mechanics) CHAPTER 4 Motion in 2D and 3D

2 Slide 1 Revision : 2. Displacement vector ( r): 1. Position vector (r): r t = x t i + y t j + z(t)k Particle s motion in 2D Position vector 1 Position vector 2

3 Slide 2 Revision : 4. Instantaneous Velocity (Velocity) (v): 3. Average Velocity (v avg ): direction of v : along the tangent to path direction of v avg = direction of r

4 Slide 3 Revision : 6. Instantaneous Acceleration (Acceleration) (a): 5. Average Acceleration (a avg ): direction of a : not related to path

5 Slide 4 Objectives covered in this lesson : 1. to identify the launched angle of a projectile that is measured from the horizontal. 2. to resolve the initial velocity of the projectile into its components and write it in unit-vector notation. 3. to analyze the projectile motion into two one-dimensional independent motions: horizontal and vertical. 4. to identify the horizontal and vertical components of the acceleration of the projectile. 5. to calculate the horizontal and vertical components of the final velocity of the projectile after time t. 6. to calculate the horizontal and vertical displacement of the projectile after time t. 7. to calculate the maximum height that the projectile can reach. 8. to calculate the time that the projectile spend to reach any position. 9. to define the horizontal Range of the projectile. 10. to calculate the horizontal Range of the projectile. 11. to calculate the maximum horizontal Range of the projectile. 12. to describe the path of the projectile (trajectory). Motion in 2D and 3D: Projectile Motion

6 Slide Projectile Motion : It s a 2D motion

7 Slide Projectile Motion : Q: can we consider the following as projectile motions: 1. a tennis ball in flight. 2. a plane in flight. 3. a duck in flight. The images show Types of Projectile Motions.

8 Slide Projectile Motion : In studying the projectile motion in this course, we assume that air has no effect on the projectile. y Max. height Projectile Projectile s Path trajectory Launching point θ 0 R The range R x Landing point

9 Slide Projectile Motion : Initial Velocity (v o ) : Scalar components: Position vector r of the motion & Velocity vector v of the motion change continuously v o is the magnitude of v o. θ o is the angle between v o and the positive x direction. Acceleration vector a of the motion is constant and is always directed downwards

10 Slide Projectile Motion : Position vector r Velocity vector v y y v y v v y y y v 0y v 0 v x v y v x v x x x x y θ 0 v 0x x

11 Slide Projectile Motion : Divide it into two 1D motions: One in the x-axis (horizontal motion) One in the y-axis (vertical motion) and study them separately.

12 Slide Projectile Motion Analyzed : Projectile Motion Horizontal Motion (x-axis) No acceleration (a x = 0) Vertical Motion (y-axis) Free-fall motion (a y = constant = g) v x = v ox x x o = v ox t v ox = v o cosθ o Substitute: a a x (0) v o v ox v o cosθ o v v x Substitute: x y a a y ( g) v o v oy v o sinθ o v v y

13 Slide Projectile Motion Analyzed :

14 Slide Projectile Motion Analyzed : the motion in x Note: When solving problems: always put the origin of the xy-graph at the start of the motion, so that x o = 0, y o = 0. the motion in y substitute x o = 0, y o = 0 then, combine by eliminating ( t )

15 Slide Projectile Motion Analyzed : The Range (R): is the horizontal distance the projectile has travelled when it returns to its initial (launch) height. the motion in x the motion in y substitute x x o = R, y y o = 0 then, combine by eliminating ( t ) and using sin 2θ = 2 sin θ cos θ

16 Slide Projectile Motion Analyzed : Note: If the projectile s final height is not the same as its initial height: The horizontal range the horizontal distance travelled The maximum horizontal range the maximum horizontal distance travelled y R Horizontal distance x R Horizontal distance

17 Slide Projectile Motion Analyzed : Maximum Height: Max. height H x

18 Slide Projectile Motion Analyzed : Answer: (a) v x is constant. (b) v y is initially positive, decreases to zero, and then becomes progressively more negative. (c) a x = 0 throughout the motion. (d) a y = g throughout the motion.

19 Slide 18 Problem 21 :

20 Slide 19 Problem 21 :

21 Slide 20 Problem 38 :

22 Slide 21 Problem 38 :

23 Slide 22 Sample Problem (4-7) :

24 Slide 23 Sample Problem (4-7) :

25 Chapter 4 : Vectors Slide 24 (last) Summary: Motion in 2D and 3D: Projectile Motion. Projectile Motion Analyzed. Next lesson we will cover: Section (4-7). Sample problem (4-10). Any Questions?

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