motion in a plane - oduww2.odu.edu/~jdudek/phys111n_materials/2_2d_motion.pdf · physics 111n 26...

physics 111N

motion in a plane

physics 111N 2

position & displacement vectors

we’re plotting the plane (e.g. billiard table viewed from above)

! the position vector points from the origin to the object

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physics 111N 3

position & displacement vectors

t�2.83 s

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y�m

t�4.24 s

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easier to visualize as

! the displacement vector points from where the object was to where it is now

physics 111N 4

position & displacement vectors - components

t�2.83 s

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y�m

t�4.24 s

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physics 111N 5

average velocity

t�4.24 s

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y�m! the average velocity vector is defined to be the rate of change of displacement vector

! the components of the average velocity vector are the rate of change of the components of the displacement vector

physics 111N 6

instantaneous velocity

! the instantaneous velocity vector is defined analogously to one-dimension

physics 111N 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 8

instantaneous velocity

! the instantaneous velocity vector can be expressed via components

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y�m

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physics 111N 9

dragonfly

Find the x & y components of the average velocity between A & BFind the magnitude and direction of the average velocity between A & BIndicate the direction of the instantaneous velocity at A and B on the diagram

A dragonfly follows the path shown, moving from point A to point B in 1.50 s.

physics 111N 10

acceleration in a plane

! the definitions of average and instantaneous acceleration are the obvious extensions

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y�m

physics 111N 11

acceleration in a plane

! the definitions of average and instantaneous acceleration are the obvious extensions

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y�m

physics 111N 12

acceleration in a plane

! acceleration parallel to the velocity changes the magnitude of velocity but not the direction

physics 111N 13

acceleration in a plane

! acceleration perpendicular to the velocity changes the direction of velocity but not the magnitude

physics 111N 14

acceleration in a plane

! a general acceleration vector can change both the magnitude and direction of the velocity

physics 111N 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 17

projectile motion

we’d like to to be able to describe why the ball takes this path

physics 111N 18

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 19

projectile motion

! our theory of projectile motion is that

! and the constant acceleration equations apply to x & y separately

physics 111N 20

projectile motion

! our theory of projection motion is that

! and the constant acceleration equations apply to x & y separately

physics 111N 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

note that it’s very similar to the kicked football

physics 111N 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 & the acceleration

physics 111N 23

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 24

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 25

projectile motion

! we can analyse this motion using our equations

! at t=0

physics 111N 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 .

Find the horizontal range, R, the horizontal distance travelled before hitting the ground

physics 111N

h

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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 .

at the highest point

physics 111N

h

t�1.44 s

10 20 30 40 x�m

�10

�5

5

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y�m

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 time when the ball reaches the highest point of its flight and that height, h .

at the highest point

physics 111N 29

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

R

t�2.89 s

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y�mFind the horizontal range, R, the horizontal distance travelled before hitting the ground

ball hits the ground when

hmmm, twice the time to reach the apex ?

physics 111N 34

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

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35

uniform circular motion

the triangles are ‘similar’