Chapter 4 Two-Dimensional Kinematics. Copyright 2010 Pearson Education, Inc.

Transcription

1 Chapter 4 Two-Dimensional Kinematics

2 Units of Chapter 4 Motion in Two Dimensions Projectile Motion: Basic Equations Zero Launch Angle General Launch Angle Projectile Motion: Key Characteristics

3 4-1 Motion in Two Dimensions If elocity is constant, motion is along a straight line:

4 4-1 Motion in Two Dimensions Motion in the x- and y-directions should be soled separately:

5 4-2 Projectile Motion: Basic Equations Assumptions: ignore air resistance g 9.81 m/s 2, downward ignore Earth s rotation If y-axis points upward, acceleration in x-direction is zero and acceleration in y-direction is m/s 2

6 What is the motion of a struck baseball? Once it leaes the bat (if air resistance is negligible) only the force of graity acts on the baseball.

7 The baseball has a x 0 and a y g, it moes with constant elocity along the x-axis and with nonzero, constant acceleration along the y-axis.

8 Example: An object is projected from the origin. The initial elocity components are ix 7.07 m/s, and iy 7.07 m/s. Determine the x and y position of the object at 0.2 second interals for 1.4 seconds. Also plot the results.! y! x iy ix! t! t a y! t 2 Since the object starts from the origin, Δy and Δx will represent the location of the object at time Δt.

9 Example continued: t (sec) x (meters) y (meters)

10 4-2 Projectile Motion: Basic Equations The acceleration is independent of the direction of the elocity:

11 4-2 Projectile Motion: Basic Equations These, then, are the basic equations of projectile motion:

12 4-3 Zero Launch Angle Launch angle: direction of initial elocity with respect to horizontal

13 4-3 Zero Launch Angle In this case, the initial elocity in the y-direction is zero. Here are the equations of motion, with x 0 0 and y 0 h:

14 4-3 Zero Launch Angle This is the trajectory of a projectile launched horizontally:

15 4-3 Zero Launch Angle Eliminating t and soling for y as a function of x: This has the form y a + bx 2, which is the equation of a parabola. The landing point can be found by setting y 0 and soling for x:

16 4-4 General Launch Angle In general, 0x 0 cos θ and 0y 0 sin θ This gies the equations of motion:

17 4-4 General Launch Angle Snapshots of a trajectory; red dots are at t 1 s, t 2 s, and t 3 s

18 4-5 Projectile Motion: Key Characteristics Range: the horizontal distance a projectile traels If the initial and final eleation are the same:

19 4-5 Projectile Motion: Key Characteristics The range is a maximum when θ 45 :

20 4-5 Projectile Motion: Key Characteristics Symmetry in projectile motion:

21 Example (text problem 3.50): An arrow is shot into the air with θ 60 and i 20.0 m/s. (a) What are x and y of the arrow when t 3 sec? y i The components of the initial elocity are: 60 x ix iy i i cos! 10.0 m/s sin! 17.3 m/s At t 3 sec: fx fy ix iy + + a a x y " t " t ix iy 10.0 m/s! g" t! 12.1 m/s

22 (b) What are the x and y components of the displacement of the arrow during the 3.0 sec interal? y x r 7.80 m m ! "!! +!!! +!! +!!! t g t t a t y r t t a t x r iy y iy y ix x ix x Example continued:

23 Example: How far does the arrow in the preious example land from where it is released? 1 The arrow lands when Δy 0.! y! "! 2 iy t g t 0 2 Soling for Δt:! t 2 g iy 3.53sec The distance traeled is:! x ix ix! t! t a! t 2 x m

24 Summary of Chapter 4 Components of motion in the x- and y- directions can be treated independently In projectile motion, the acceleration is g If the launch angle is zero, the initial elocity has only an x-component The path followed by a projectile is a parabola The range is the horizontal distance the projectile traels