Position update formula

tvrr

t

rr

t

rv

if

if

avg

avg

We can use this to predict the motion of an object, if we know (or can guess) vavg.

This formula is the key to simulating motion with computer programs.

tvrr if avg

模拟

1. Calculate new r2. Calculate new vavg (how?...)3. Next time step

My research: simulating the motion of proteins 蛋白质

Proteins are the molecular machines that must work to keep living things alive.

Many diseases are caused by ‘broken’ proteins.

Special case: constant acceleration

2

2

1tatvrr iif

A common example of constant acceleration: falling.

Which direction is the acceleration?

Which direction is the acceleration?

Which direction is the acceleration?

Parallel (or anti-parallel) to the velocity, changing the speed.

平行

Which direction is the acceleration?

Perpendicular to the velocity, changing the direction of motion.

垂直

Curved motion

Acceleration can be split into two types:

(1) Acceleration that changes speed, and not direction

(2) Acceleration that changes direction, and not speed

Acceleration can be split into two types:

rule)(product d

ˆdˆ

d

d

ˆ d

dd

d

t

vvv

t

v

vvt

t

va

Change in speed:tangential acceleration at

Change in direction:radial acceleration ar

Let’s calculate ar .

θ R

1̂v

R

2v̂

After a time Δt, particle moves to position 2.

RR

tv

θ

θ1

1v̂

These are similar triangles.

θ R

1̂v

R

2v̂

After a time Δt, particle moves to position 2.

RR

tv

θ

θ 11

v̂

These are similar triangles.

RR

tv

θ

θ 11

v̂

These are similar triangles, so:

R

v

t

vva

R

v

t

v

v

R

tv

r

2ˆ

ˆ1

ˆ

The radial acceleration is also called centripetal acceleration.

Its magnitude is

R

var

2

向心加速度

and its direction is towards the center of the circle.

Example: The Moon’s orbit around the Earth

Assume a circular orbit. (It’s really an ellipse.)

假设

R = 3.84 x 108 m

Period of one orbit: T = 27.3 days

moonv