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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
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
∆𝑟 = 𝑟 2 − 𝑟 1
∆𝑟 = ∆𝑥𝑖 + ∆𝑦𝑗 + ∆𝑧𝑘
𝑟 1
𝑟 2
∆𝑟
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
𝑟 1
𝑟 2
∆𝑟
𝑣 𝑎𝑣𝑔 =∆𝑟
∆𝑡
𝑣 𝑎𝑣𝑔 =∆𝑥
∆𝑡𝑖 +
∆𝑦
∆𝑡𝑗 +
∆𝑧
∆𝑡𝑘
Experimental Physics - Mechanics - Motion in 2D and 3D 4
Instantaneous velocity
y
x
1
2
path
Always tangent to the path.
222
zyx vvvv
𝑟 1
𝑟 2
∆𝑟
𝑣 1
𝑣 2
𝑣 = lim∆𝑡→0
∆𝑟
∆𝑡= 𝑑𝑟
𝑑𝑡
𝑣 =𝑑𝑥
𝑑𝑡𝑖 +
𝑑𝑦
𝑑𝑡𝑗 +
𝑑𝑧
𝑑𝑡𝑘
𝑣 = 𝑣𝑥𝑖 + 𝑣𝑦𝑗 + 𝑣𝑧𝑘
Experimental Physics - Mechanics - Motion in 2D and 3D 5
Average acceleration
y
x
1
2
path
𝑟 1
𝑟 2
∆𝑟
𝑣 1
𝑣 2 𝑣 1
𝑣 2
𝑎 𝑎𝑣𝑔
∆𝑣
𝑎 𝑎𝑣𝑔 =∆𝑣
∆𝑡
𝑎 𝑎𝑣𝑔 =∆𝑣𝑥∆𝑡
𝑖 +∆𝑣𝑦
∆𝑡𝑗 +
∆𝑣𝑧∆𝑡
𝑘
Experimental Physics - Mechanics - Motion in 2D and 3D 6
Instantaneous acceleration
y
x
1
2
path
Change in either magnitude or in direction.
𝑎 = lim∆𝑡→0
∆𝑣
∆𝑡= 𝑑𝑣
𝑑𝑡=𝑑2𝑟 𝑑𝑡2
𝑎 =𝑑𝑣𝑥𝑑𝑡
𝑖 +𝑑𝑣𝑦
𝑑𝑡𝑗 +
𝑑𝑣𝑧𝑑𝑡
𝑘
𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 𝑟 1
𝑟 2
𝑣 1
𝑣 2
𝑎 1
𝑎 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 𝑣 = 𝑣𝑥𝑖 + 𝑣𝑦𝑗 + 𝑣𝑧𝑘
𝑣 = 𝑣 0 + 𝑎 𝑡
𝑟 = 𝑥𝑖 + 𝑦𝑗 + 𝑧𝑘
𝑟 = 𝑟 0 + 𝑣 𝑡 + 12𝑎𝑡2
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
𝑣 0
𝑣 𝑣
𝑣
𝑣
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
𝑟 = 𝑟 0 + 𝑣 𝑡 + 12𝑎𝑡2
𝑟
𝑣 𝑡
12𝑎𝑡2
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 – horizontal motion
tvxx x00
ga
a
y
x 0
tvxx )cos( 000
𝑣 0 𝑟
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 – vertical motion
2/2
00 gttvyy y
ga
a
y
x 0
gtvvy 00 sin
)(2)sin( 0
2
00
2 yygvvy
𝑣 0 𝑟
Experimental Physics - Mechanics - Motion in 2D and 3D 12
Projectile – 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
𝑣 0 𝑟
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 – h and R
g
vh
2
sin 0
22
0
00
2
0 cossin2
g
vR
gvRR /)4/( 2
00max
𝑣 0
𝑣 𝑣
𝑣
Experimental Physics - Mechanics - Motion in 2D and 3D 14
Projectile – the longest range R
0 5 10 150
2
4
6
8
75°
60°
45°
30°
y(m
)
x(m)
v0 = 12 m/s
15°
Experimental Physics - Mechanics - Motion in 2D and 3D 15
Circular uniform motion
r
∆𝑣
𝑣 1 𝑣 2
𝑣 1
𝑣 2
𝜆 = 2𝜃𝑟
∆𝑡 =𝜆
𝑣=
2𝑟𝜃
𝑣
𝑣 2 = 𝑣 cos 𝜃𝑖 − 𝑣 sin 𝜃𝑗
𝑣 1 = 𝑣 cos 𝜃𝑖 + 𝑣 sin 𝜃𝑗
∆𝑣 = −2𝑣 sin 𝜃𝑗
𝑎 =∆𝑣
∆𝑡=
𝑣2 sin 𝜃
𝑟𝜃(−𝑟 )
𝑎𝑟 =𝑣2
𝑟
Experimental Physics - Mechanics - Motion in 2D and 3D 16
Tangential and radial accelerations
r
𝑎 𝑟 =𝑣2
𝑟(−𝑟 ) 𝑎 𝑡 =
𝑑𝑣
𝑑𝑡𝑡
∆𝑣
𝑣 1
𝑣 2 ∆𝑣 𝑡
∆𝑣 𝑟
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
𝑣 𝑚 = 𝑣 𝑡 + 𝑉𝑚
𝑎 𝑚 = 𝐴 𝑚
𝑅𝑚0 𝑅𝑚𝑓
𝑟 𝑚0 𝑟 𝑡0 𝑟 𝑡𝑓
𝑟 𝑚𝑓
Experimental Physics - Mechanics - Motion in 2D and 3D 18
Equations of motion in the vector form.
Projectile motion – 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!
