Motion in Two Dimensions

1. Motionin Two Dimensions Chapter 7.2

2. Projectile Motion • What is the path of a projectile as it moves through the air? • ? • ? • . • What forces act on projectiles? • Only , which acts only in the y-direction. • is ignored in projectile motion.

3. Choosing Coordinates & Strategy • For projectile motion: • Choose the for motion where is a factor. • Choose the for motion. Since there are forces acting in this direction (of course we will neglect friction due to air resistance), the speed will be (a = __). • Analyze motion along the y-axis from the x-axis. • If you solve for in one direction, you automatically solve for in the other direction.

4. The Trajectory of a Projectile • What does the free-body diagram look like for force?

5. ay = The Vectors of Projectile Motion • What vectors exist in projectile motion? • in both the ___ and ___ directions. • in the ___ direction only. vx () • Why is the velocity constant in the x-direction? • . ax = vy () • Why does the velocity increase in the y-direction?

6. Ex. 1: Launching a Projectile Horizontally • A cannonball is shot horizontally off a cliff with an initial velocity of 30 m/s. If the height of the cliff is 50 m: • How far from the base of the cliff does the cannonball hit the ground? • With what speed does the cannonball hit the ground?

7. a = ___ ___m vx vf = ? vy x = ? Diagram the problem vi = Fg = Fnet

8. State the Known & Unknown • Known: • xi = 0 • vix = ___ m/s • yi = ___ m/s • viy = ___ m/s • a = ___ = ___ m/s2 • y = ____ m • Unknown: • x at y = -50 m • vf = ?

9. Perform Calculations (y) • Strategy: • Use reference table to find formulas you can use. • vfy = • y = • Note that __ has been substituted for __ and __ for __. • Use known factors such as in this case where the initial velocity in the y-direction is known to be to simplify the formulas. • vfy = • y =

10. Perform Calculations (y) • Now that we have , we can use the first formula to find the final velocity. • vfy = • vy =

11. Perform Calculations (x) • Strategy: • Since you know the for the (), you also have it for the . • is the only variable that can transition between motion in both the __ and __ directions. • Since we air resistance and gravity does not act in the (x-direction), a = __. • Choose a formula from your reference table • x = • Since a = ___, the formula reduces to x = _____ • x =

12. Finding the Final Velocity (vf) • We were given the initial x-component of velocity, and we calculated the y-component at the moment of impact. • Logic: Since there is no acceleration in the horizontal direction, then ____ = ____ • We will use the . vfx = vf = ? vfy =

13. Ex. 2: Projectile Motion above the Horizontal • A ball is thrown from the top of the Science Wing with a velocity of 15 m/s at an angle of 50 degrees above the horizontal. • What are the x and y components of the initial velocity? • What is the ball’s maximum height? • If the height of the Science wing is 12 m, where will the ball land?

14. y x vi = _____ viy ay =  = ____° vi = ____ m/s vix  = ___° ___ m Ground x = ? Diagram the problem

15. State the Known & Unknown • Known: • xi = 0 • yi = ___ m • vi = ___ m/s •  = ___° • a = ___ = ____ • Unknown: • ymax = ? • t = ? • x = ? • viy = ? • vix = ?

16.  = 50° vi = 15 m/s vyi vxi Perform the Calculations (ymax) • y-direction: • Initial velocity: viy = • viy = • viy = • Time when vfy = 0 m/s: vfy = viy – gt (ball at ) • t = • t = • t = • Determine the maximum height: ymax = • ymax = • ymax =

17. Perform the Calculations (t) • Since the ball will accelerate due to gravity over the distance it is falling back to the ground, the time for this segment can be determined as follows • Time from peak to when ball hits the ground: • From reference table: ymax = • Since ___ can be set to zero as can ___, • t = • t = • t = • By adding the it takes the ball to reach its maximum height ()to the it takes to reach the ground will give you the total . • ttotal =

18.  = 50° vi = 15 m/s vyi vxi Perform the Calculations (x) • x-direction: • Initial velocity: • vix = • vix = • Determine the total distance: • x = • x =

19. Analyzing Motion in the x and y directions independently. • x-direction: • dx = vix t = vfxt • vix = vicos • y-direction: • dy = ½ (vi + vf) t = vavg t • vf = viy + gt • dy = viy t + ½ g(t)2 • vfy2 = viy2 + 2gd • viy = visin

20. Key Ideas • Projectile Motion: • is the only force acting on a projectile. • Choose a coordinate axis that where the x-direction is along the and the y-direction is . • Solve the x and y components . • If is found for one dimension, it is also known for the other dimension.