Curvilinear Motion

1. Curvilinear Motion • Motion of projectile • Normal and tangential components

2. Projectile motion A kicker should know at what angle, q, and initial velocity vo, he must kick the ball to make a field goal. For a given kick “strength”, at what angle should the ball be kicked to get the maximum distance?

3. A fireman wishes to know the maximum height on the wall he can project water from the hose. At what angle, q, should he hold the hose?

5. Projectile motion Horizontal – zero acceleration Vertical – constant acceleration ( i.e, gravity) The horizontal distance between the yellow ball is constant since the velocity in the horizontal direction is constant.

6. KINEMATIC EQUATIONS: HORIZONTAL MOTION

7. v = vo + act s = so + vot + act2 v2 = vo2 + 2ac(s-so) v = vo + act s = so + vot + act2 v2 = vo2 + 2ac(s-so) vy = (vo)y - gt y = yo + (vo)yt - gt2 vy2 = (vo)y2 – 2g(y-yo) Horizontal vx = (vo)x x = xo + (vo)xt vx = (vo)x Vertical

8. Example 1 A sack slides off the ramp, with a horizontal velocity of 12 m/s. If the height of the ramp is 6m from the floor, determine (i) the time needed for the sack to strike the floor (ii) the range R where sacks begin to pile up

9. solution (vA)x = 12 m/s (vA)y = 0 (vB)x = (vA)x = 12 m/s Unknowns : (vb)y time of flight toA R Vertical motion : Horizontal motion :

10. Example 2 The chipping machine is designed to eject wood at vo= 7.5 m/s as shown. If the tube is oriental at 30o from the horizontal, determine how high h, the pile if they land on the pile 6m from the tube

11. solution (va)x = (vo)x Unknowns : height h time of flight toA vertical component of velocity (va)y

12. Horizontal Vertical

13. Curvilenear motion Normal and tangential components

14. \ v=0.12t2 When t=0, s=0 When t=10, s=? when t = 0, v =0 → c=0 s =0.6(102) = 60 m when t = 10,

15. Curvilenear motion Normal and tangential motion components Cars traveling along a clover-leaf interchange experience an acceleration due to a change in speed as well as due to a change in direction of the velocity. If the car’s speed is increasing at a known rate as it travels along a curve, how can we determine the magnitude and direction of its total acceleration? Why would you care about the total acceleration of the car?

16. A motorcycle travels up a hill for which the path can be approximated by a function y = f(x). If the motorcycle starts from rest and increases its speed at a constant rate, how can we determine its velocity and acceleration at the top of the hill? How would you analyze the motorcycle's “flight” at the top of the hill?

17. ACCELERATION IN THE n-t COORDINATE SYSTEM There are two components to the acceleration vector: normal component tangential component

18. Constant acceleration

19. tangential components • tangent to the curve and in the direction of increasing or decreasing velocity. • represents the time rate of change in the magnitude of the velocity Normal/centripetal components • Always directed toward the center of • curvature of the curve. • represents the time rate of change in • the direction of the velocity

20. Magnitude of acceleration Radius of curvature

21. Consider these 2 cases: • If the particle moves along a straight line • If the particle moves along a curve with a constant speed

23. Example 2 when t = 10, v =100 The boxes travel along the industrial conveyor. If a box starts from rest at A and increases its speed such that at = (0.2t) m/s2, determine the magnitude of its acceleration when it arrives at point B. when t = t’, v =0

24. magnitude of acceleration at B; info we have; [solution]

25. distance from A to B; to find tB

26. At point B;