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Experimental Physics
EP1 MECHANICS
- Motion in 2D and 3D -
Rustem Valiullin
http://uni-leipzig.de/~valiu/
Experimental Physics - Mechanics - Motion in 2D and 3D
1
Position and Displacement

r  xˆi  yˆj  zkˆ
y
  
r  r2  r1
1

r1

r  xˆi  yˆj  zkˆ

r
2

r2
path
x
Experimental Physics - Mechanics - Motion in 2D and 3D
2
Average velocity

vavg
y
1

r1

x ˆ y ˆ z ˆ
vavg 
i
j
k
t
t
t

r
The average velocity
is not a function of
path connecting
two points.
2

r2

r

t
path
x
Experimental Physics - Mechanics - Motion in 2D and 3D
3
Instantaneous velocity



r dr
v  lim

t 0 t
dt
y

v1
1

r1
Always tangent to the path.
 dx ˆ dy ˆ dz ˆ
v
i
j k
dt
dt
dt

v  vx ˆi  v y ˆj  vz kˆ

r
2

r2

v2
path
v  vx2  v 2y  vz2
x
Experimental Physics - Mechanics - Motion in 2D and 3D
4
Average acceleration

aavg
y

v1
1

r1

vx ˆ v y ˆ vz ˆ
aavg 
i
j
k
t
t
t

r

aavg
2

r2

v

t

v2

v
path

v1

v2
x
Experimental Physics - Mechanics - Motion in 2D and 3D
5
Instantaneous acceleration


2

v dv d r
a  lim

 2
t 0 t
dt dt
y

v1
1
Change in either magnitude
or in direction.

a1

r1
2

r2

v2

a2
 dvx ˆ dv y ˆ dvz ˆ
a
i
j
k
dt
dt
dt

a  ax ˆi  a y ˆj  az kˆ
path
x
Experimental Physics - Mechanics - Motion in 2D and 3D
6
Constant acceleration

v  vx ˆi  v y ˆj  vz kˆ
v x  v x 0  a x t

v y  v y 0  a y t

v z  v z 0  a z t
  
v  v0  at
1 2

 x  x0  v x 0t  2 a x t

1 2

 y  y0  v y 0t  a y t
2

1 2

 z  z0  vz 0t  2 a z t

r  xˆi  yˆj  zkˆ
  
12
r  r0  v0t  at
2
Experimental Physics - Mechanics - Motion in 2D and 3D
7
Projectile motion
y
vy=0
10

v
vy

v0
5
v0y

v
vx
vy
h
vx

v

0
0
v0x
5
10
15
20
25
vx
R
-5
x
30
vy

v
-10
Experimental Physics - Mechanics - Motion in 2D and 3D
8
Projectile motion

gt2 / 2
y

v0 t
10

v0
5
v0y

0
0
-5
-10

r
v0x
5
10
15
20
x
  
12
r  r0  v0t  gt
2
Experimental Physics - Mechanics - Motion in 2D and 3D
9
Projectile – horizontal motion
y
y
10

v0
5
v0y
0
-10
x

0
-5

r
v0x
5
10
a x  0

a y   g
15
20
x
x  x0  v0 xt
x  x0  (v0 cos 0 )t
Experimental Physics - Mechanics - Motion in 2D and 3D
10
Projectile – vertical motion
y
y
10

v0
5
v0y
0
-10
x

0
-5

r
v0x
5
10
a x  0

a y   g
15
20
x
y  y0  v0 y t  gt 2 / 2
v y  v0 sin 0  gt
v  (v0 sin 0 )  2 g ( y  y0 )
2
y
Experimental Physics - Mechanics - Motion in 2D and 3D
2
11
Projectile – equation of path
y
y
10

v0
5
v0y
0
-10
x

0
-5
y  f (x)

r
v0x
5
10
x  x0  (v0 cos 0 )t
15
20
x
y  y0  (v0 sin 0 )t  gt 2 / 2
g
2
y  (tan 0 ) x 
x
2(v0 cos 0 )2
Experimental Physics - Mechanics - Motion in 2D and 3D
12
Projectile – h and R
y
vy=0
10

v
vy

v0
5
v0y

v
vx
vy
h
vx

v

0
0
v0x
5
10
15
20
25
30
x
R
-5
-10
v02 sin 2 0
h
2g
2
0
2v
R
sin 0 cos 0
g
Rmax  R(0   / 4)  v02 / g
Experimental Physics - Mechanics - Motion in 2D and 3D
13
Projectile – the longest range R
8
v0 = 12 m/s
75°
6
y(m)
60°
4
45°
2
30°
15°
0
0
5
10
15
x(m)
Experimental Physics - Mechanics - Motion in 2D and 3D
14
Circular uniform motion


v1

v2

v

v2
ar 
r
  2r
t 

v

r
2r
v


v2  v cos ˆi  v sin ˆj

ˆ
ˆ

v1  v cos i  v sin j

 v v 2 sin 
a

( rˆ)
t
r

v  2v sin  (ˆj)
Experimental Physics - Mechanics - Motion in 2D and 3D

v1

v2


15
Tangential and radial accelerations

v1

vr

vt

v

v2

r

 dvr v 2
ar 
 (  rˆ )
dt
r

 dv
at  t
dt
Experimental Physics - Mechanics - Motion in 2D and 3D
16
Relative motion

Rmf

Rm 0

rm 0

rt 0

rtf

rmf
As seen by the external observer:

 
vm  vt  Vm
If the train is moving with a constant velocity,
but the man is moving with an acceleration in the train
Experimental Physics - Mechanics - Motion in 2D and 3D


am  Am
17
To remember!
 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 tengential acceleration.
 There are simple rules relating velocities
and accelerations in two reference systems
moving with respect to each other.
Experimental Physics - Mechanics - Motion in 2D and 3D
18
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