Mathematical ModellingMathematical ModellingMathematical ModellingMathematical Modelling

Name: Ko-Huang Kong TatName: Ko-Huang Kong Tat

Class: F.6CClass: F.6C

Class Number:16Class Number:16

Contents: Mathematical Modelling in Dynamics Throu

gh Ordinary Differential Equations of First Order

Mathematical Modelling Of Geometrical PrMathematical Modelling Of Geometrical Problems Through Ordinary Differential oblems Through Ordinary Differential Equations Of First Order Equations Of First Order

Here a particle moves in a straight line in such a manner that its acceleration is always proportional to its distance from the origin and is always directed toward the origin, so that

vdv/dx=-cx

Let a particle travel a distance x in time t in a straight line, then its velocity v is given by dx/dt and its acceleration is given by

dv/dt=(dv/dx)(dx/dt)=vdv/dx=d2x/dt2

1.1 Simple Harmonic Motion

Integrating

v2=ca2-x2

where the particle is initially at rest at x=a .Equation gives

dx/dt=- ( c )1/2 (a2-x2 )1/2

We take the negative sign since velocity increases as x decreases

Integrating again and using the condition that at t=0,x=a

x( t )=a cos(c )1/2 t

so that

v( t )=-a( c )1/2 sin ( c )1/2 t

Thus in simple harmonic motion, both displacement and

velocity are periodic functions with period 2/ ( c )1/2

The particle starts from A with zero velocity and moves

towards 0 with increasing velocity and reaches 0 at time /2( c )1/2 with

velocity ( c )1/2a.

It continue to move in the same direction,but now with decreasing

velocity till it reaches A’(OA’=a ) where its velocity is again zero.

It then begins moving towards 0 with increasing velocity and

reaches 0 with velocity ( c )1/2 a and again comes to rest at A after a

total time period2/ ( c )1/2..The periodic motion then repeats itself.

As one example of SHM,consider a particle of mass m

attaches to one end of a perfectly elastic string,the

other end of which is attaches to a

fixed point o( figure 1) .The particle

moves under gravity in vacuum.

Let Lo be the natural length of the

string and let a be its extension when

the particle is in equilibrium so that

by Hook’s law --- mg=To=a/Lo

where is the coefficient of elasticity.

Now let the string be further stretched

a distance D and then the mass be left free.The equation of

motion which states that mass x acceleration in any

direction=force on the particle in that direction, gives

mv dv/dx=mg-T=mg-( ax)/Lo=- x/Lo

or

v dv/dx= (/m) (x/Lo)=-gx/a,

which gives a simple harmonic motion with time period 2 (a/g)1/2

A particle falls under gravity in a medium in which the resistance is proportional to the velocity. The equation of motion is

1.2 Motion Under Gravity in a Resisting Medium

m dv/di=mg-mkv

Or

dv/V-v=k dt ; V=g/k

Integrating

V-v=V e-kt

If the particle starts from rest with zero velocity. Equation gives

v=V(1- e-kt)

So that the velocity goes on increasing and appproaches the limiting velocity g/k as t .Replacing v by dx/dt, we get

dx/dt=V (1- e-kt)

Integrating and using x=0 when t=0,we get

x=Vt V e-kt /k-v/k

Many geometrical entities can be expressed in terms of derivatives and as such relations between these entities can give rise to differential equations whose solution will give us a family of curves for which the given relation between geometrical entities is satisfied.

2.1 Simple Geometrical Problems