physics 111N motion in a plane

Transcription

1 physics 111N motion in a plane

2 position & displacement vectors ym! the position vector points from the origin to the object t2.83 s xm we re plotting the plane (e.g. billiard table viewed from above) physics 111N 2

3 position & displacement vectors ym ym t2.83 s t4.24 s xm xm! the displacement vector points from where the object was to where it is now easier to visualize as physics 111N 3

4 position & displacement vectors - components ym ym t2.83 s t4.24 s xm xm physics 111N 4

5 average velocity ym 15 t4.24 s! the average velocity vector is defined to be the rate of change of displacement vector xm! the components of the average velocity vector are the rate of change of the components of the displacement vector physics 111N 5

6 instantaneous velocity! the instantaneous velocity vector is defined analogously to one-dimension physics 111N 6

7 instantaneous velocity! the instantaneous velocity vector is defined analogously to one-dimension! instantaneous velocity vector points along the tangent of the x-y path physics 111N 7

8 instantaneous velocity! the instantaneous velocity vector can be expressed via components ym xm physics 111N 8

9 dragonfly A dragonfly follows the path shown, moving from point A to point B in 1.50 s. Find the x & y components of the average velocity between A & B Find the magnitude and direction of the average velocity between A & B Indicate the direction of the instantaneous velocity at A and B on the diagram physics 111N 9

10 acceleration in a plane! the definitions of average and instantaneous acceleration are the obvious extensions ym xm physics 111N 10

11 acceleration in a plane! the definitions of average and instantaneous acceleration are the obvious extensions ym xm physics 111N 11

12 acceleration in a plane! acceleration parallel to the velocity changes the magnitude of velocity but not the direction physics 111N 12

13 acceleration in a plane! acceleration perpendicular to the velocity changes the direction of velocity but not the magnitude physics 111N 13

14 acceleration in a plane! a general acceleration vector can change both the magnitude and direction of the velocity physics 111N 14

15 acceleration in a plane! a general acceleration vector can change both the magnitude and direction of the velocity! but we can understand the effect by breaking into components parallel and perpendicular to the velocity physics 111N 15

16 projectile motion we d like to to be able to describe why the ball takes this path physics 111N 17

17 projectile motion - independence of horiz. & vertical motion! another experimental observation is useful notice that the vertical motion is identical not affected by the horizontal motion great news! we can consider x & y separately! the y-motion is just free-fall! the x-motion a constant velocity physics 111N 18

18 projectile motion! our theory of projectile motion is that! and the constant acceleration equations apply to x & y separately physics 111N 19

19 projectile motion! our theory of projection motion is that! and the constant acceleration equations apply to x & y separately physics 111N 20

20 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal note that it s very similar to the kicked football physics 111N 21

21 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal show the velocity vector & the acceleration physics 111N 22

22 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal show the velocity vector components physics 111N 23

23 projectile motion! we can analyse this motion using our equations! prove that the shape of the path is a parabola eliminate t from the equations to give y as a function of x (choose x0=0, y0=0 for simplicity) physics 111N 24

24 projectile motion! we can analyse this motion using our equations! at t=0 physics 111N 25

25 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal Find the time when the ball reaches the highest point of its flight and that height, h. Find the horizontal range, R, the horizontal distance travelled before hitting the ground physics 111N 26

26 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal Find the time when the ball reaches the highest point of its flight and that height, h. ym 10 t1.44 s at the highest point 5 h xm 5 10 physics 111N 27

27 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal Find the time when the ball reaches the highest point of its flight and that height, h. ym 10 t1.44 s at the highest point 5 h xm 5 10 physics 111N 28

28 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with an initial speed of 20.0 m/s at an angle of 45 to the horizontal Find the horizontal range, R, the horizontal distance travelled before hitting the ground ym 10 t2.89 s ball hits the ground when R xm 10 hmmm, twice the time to reach the apex? physics 111N 29

29 uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would change) the velocity is clearly tangent to the circle (or it would move off the circle) hence the acceleration points always toward the center of the circle - centripetal acceleration physics 111N 34

30 uniform circular motion the triangles are similar physics 111N 35