Experimental Physics EP1 MECHANICS - Motion in 2D and 3D

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!