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physics 111N
motion in a plane
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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
t�2.83 s
0 5 10 15 x�m
5
10
15
y�m
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physics 111N 3
position & displacement vectors
t�2.83 s
0 5 10 15 x�m
5
10
15
y�m
t�4.24 s
0 5 10 15 x�m
5
10
15
y�m
easier to visualize as
! the displacement vector points from where the object was to where it is now
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physics 111N 4
position & displacement vectors - components
t�2.83 s
0 5 10 15 x�m
5
10
15
y�m
t�4.24 s
0 5 10 15 x�m
5
10
15
y�m
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physics 111N 5
average velocity
t�4.24 s
0 5 10 15 x�m
5
10
15
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
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physics 111N 6
instantaneous velocity
! the instantaneous velocity vector is defined analogously to one-dimension
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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
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physics 111N 8
instantaneous velocity
! the instantaneous velocity vector can be expressed via components
0 5 10 15 x�m5
10
15
y�m
0 5 10 15 x�m5
10
15
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.
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physics 111N 10
acceleration in a plane
! the definitions of average and instantaneous acceleration are the obvious extensions
0 5 10 15 x�m5
10
15
y�m
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physics 111N 11
acceleration in a plane
! the definitions of average and instantaneous acceleration are the obvious extensions
0 5 10 15 x�m5
10
15
y�m
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physics 111N 12
acceleration in a plane
! acceleration parallel to the velocity changes the magnitude of velocity but not the direction
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physics 111N 13
acceleration in a plane
! acceleration perpendicular to the velocity changes the direction of velocity but not the magnitude
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physics 111N 14
acceleration in a plane
! a general acceleration vector can change both the magnitude and direction of the velocity
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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
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physics 111N 17
projectile motion
we’d like to to be able to describe why the ball takes this path
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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
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physics 111N 19
projectile motion
! our theory of projectile motion is that
! and the constant acceleration equations apply to x & y separately
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physics 111N 20
projectile motion
! our theory of projection motion is that
! and the constant acceleration equations apply to x & y separately
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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
![Page 21: motion in a plane - ODUww2.odu.edu/~jdudek/Phys111N_materials/2_2D_motion.pdf · physics 111N 26 projectile motion! e.g. a cannonball fired from ground level (y=0m) at x=0m with](https://reader031.vdocument.in/reader031/viewer/2022022607/5b84d58c7f8b9ae5498d1b3a/html5/thumbnails/21.jpg)
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
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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
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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)
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physics 111N 25
projectile motion
! we can analyse this motion using our equations
! at t=0
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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
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physics 111N
h
t�1.44 s
10 20 30 40 x�m
�10
�5
5
10
y�m
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
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physics 111N
h
t�1.44 s
10 20 30 40 x�m
�10
�5
5
10
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
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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
10 20 30 40 x�m
�10
�5
5
10
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 ?
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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”
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physics 111N
0 5 10 15 x�m
5
10
15
y�m
35
uniform circular motion
the triangles are ‘similar’