experimental physics ep1 mechanics - motion in 2d and 3d
TRANSCRIPT
Experimental Physics - Mechanics - Motion in 2D and 3D 1
Experimental Physics
EP1 MECHANICS
- Motion in 2D and 3D -
Rustem Valiullin
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics - Mechanics - Motion in 2D and 3D 2
Position and Displacement
y
x
1
2
path
π = π₯π + π¦π + π§π
βπ = π 2 β π 1
βπ = βπ₯π + βπ¦π + βπ§π
π 1
π 2
βπ
Experimental Physics - Mechanics - Motion in 2D and 3D 3
Average velocity
The average velocity
is not a function of
path connecting
two points.
y
x
1
2
path
π 1
π 2
βπ
π£ ππ£π =βπ
βπ‘
π£ ππ£π =βπ₯
βπ‘π +
βπ¦
βπ‘π +
βπ§
βπ‘π
Experimental Physics - Mechanics - Motion in 2D and 3D 4
Instantaneous velocity
y
x
1
2
path
Always tangent to the path.
222
zyx vvvv
π 1
π 2
βπ
π£ 1
π£ 2
π£ = limβπ‘β0
βπ
βπ‘= ππ
ππ‘
π£ =ππ₯
ππ‘π +
ππ¦
ππ‘π +
ππ§
ππ‘π
π£ = π£π₯π + π£π¦π + π£π§π
Experimental Physics - Mechanics - Motion in 2D and 3D 5
Average acceleration
y
x
1
2
path
π 1
π 2
βπ
π£ 1
π£ 2 π£ 1
π£ 2
π ππ£π
βπ£
π ππ£π =βπ£
βπ‘
π ππ£π =βπ£π₯βπ‘
π +βπ£π¦
βπ‘π +
βπ£π§βπ‘
π
Experimental Physics - Mechanics - Motion in 2D and 3D 6
Instantaneous acceleration
y
x
1
2
path
Change in either magnitude or in direction.
π = limβπ‘β0
βπ£
βπ‘= ππ£
ππ‘=π2π ππ‘2
π =ππ£π₯ππ‘
π +ππ£π¦
ππ‘π +
ππ£π§ππ‘
π
π = ππ₯π + ππ¦π + ππ§π π 1
π 2
π£ 1
π£ 2
π 1
π 2
Experimental Physics - Mechanics - Motion in 2D and 3D 7
Constant acceleration
2
00
2
00
2
00
2
1
2
1
2
1
tatvzz
tatvyy
tatvxx
zz
yy
xx
tavv
tavv
tavv
zzz
yyy
xxx
0
0
0 π£ = π£π₯π + π£π¦π + π£π§π
π£ = π£ 0 + π π‘
π = π₯π + π¦π + π§π
π = π 0 + π£ π‘ + 12ππ‘2
0 5 10 15 20 25 30
-10
-5
0
5
10
vy=0
vy
vy
vx
vxvy
vx
v0y
y
x
v0x
R
h
Experimental Physics - Mechanics - Motion in 2D and 3D 8
Projectile motion
π£ 0
π£ π£
π£
π£
Experimental Physics - Mechanics - Motion in 2D and 3D 9
0 5 10 15 20
-10
-5
0
5
10
v0y
y
x
v0x
Projectile motion
π = π 0 + π£ π‘ + 12ππ‘2
π
π£ π‘
12ππ‘2
Experimental Physics - Mechanics - Motion in 2D and 3D 10
0 5 10 15 20
-10
-5
0
5
10 y
v0y
y
x
v0x
x
Projectile β horizontal motion
tvxx x00
ga
a
y
x 0
tvxx )cos( 000
π£ 0 π
Experimental Physics - Mechanics - Motion in 2D and 3D 11
0 5 10 15 20
-10
-5
0
5
10 y
v0y
y
x
v0x
x
Projectile β vertical motion
2/2
00 gttvyy y
ga
a
y
x 0
gtvvy 00 sin
)(2)sin( 0
2
00
2 yygvvy
π£ 0 π
Experimental Physics - Mechanics - Motion in 2D and 3D 12
Projectile β equation of path
0 5 10 15 20
-10
-5
0
5
10 y
v0y
y
x
v0x
x
)(xfy
tvxx )cos( 000 2/)sin( 2
000 gttvyy
2
2
00
0)cos(2
)(tan xv
gxy
π£ 0 π
Experimental Physics - Mechanics - Motion in 2D and 3D 13
0 5 10 15 20 25 30
-10
-5
0
5
10
vy=0
vy
vxvy
vx
v0y
y
x
v0x
R
h
Projectile β h and R
g
vh
2
sin 0
22
0
00
2
0 cossin2
g
vR
gvRR /)4/( 2
00max
π£ 0
π£ π£
π£
Experimental Physics - Mechanics - Motion in 2D and 3D 14
Projectile β the longest range R
0 5 10 150
2
4
6
8
75Β°
60Β°
45Β°
30Β°
y(m
)
x(m)
v0 = 12 m/s
15Β°
Experimental Physics - Mechanics - Motion in 2D and 3D 15
Circular uniform motion
r
βπ£
π£ 1 π£ 2
π£ 1
π£ 2
π = 2ππ
βπ‘ =π
π£=
2ππ
π£
π£ 2 = π£ cos ππ β π£ sin ππ
π£ 1 = π£ cos ππ + π£ sin ππ
βπ£ = β2π£ sin ππ
π =βπ£
βπ‘=
π£2 sin π
ππ(βπ )
ππ =π£2
π
Experimental Physics - Mechanics - Motion in 2D and 3D 16
Tangential and radial accelerations
r
π π =π£2
π(βπ ) π π‘ =
ππ£
ππ‘π‘
βπ£
π£ 1
π£ 2 βπ£ π‘
βπ£ π
Experimental Physics - Mechanics - Motion in 2D and 3D 17
Relative motion
As seen by the external observer:
If the train is moving with a constant velocity,
but the man is moving with an acceleration in the train
π£ π = π£ π‘ + ππ
π π = π΄ π
π π0 π ππ
π π0 π π‘0 π π‘π
π ππ
Experimental Physics - Mechanics - Motion in 2D and 3D 18
Equations of motion in the vector form.
Projectile motion β motion in a plane with the free-fall
acceleration.
Uniform circular motion leads to centripetal
acceleration directed towards the center.
There might be a tangential acceleration.
There are simple rules relating velocities
and accelerations in two reference systems
moving with respect to each other.
To remember!