Phythagoras Theorem

Pythagoras Theorem 

Statement

" In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides"

Given

ABC is a right triangle. AC, AB and BCare hypotenuse and the other two sides respectively.

To Prove that

(AC)² = (AB)² + (BC)²

Construction

Draw BD perpendicular on AC.

Proof

Consider in right triangle ABC and ADB
<BAC = <BAD   (Common)
<ABC = < ADB  (Each right angle)
<ACB = <ABD   ( Remain angle)

Hence,
Triangle ABC and ADB are similar under AAA similarity.

Therefore,
AC/AB = AB/AD
ACxAD = (AB)²  ........  (i)

Now, consider in right triangle ABC and BDC
<BAC = <CBD   (Remain angle)
<ABC = < CDB  (Each right angle)

<ACB = <BCD   ( Common angle)

Hence,
Triangle ABC and BDC are similar under AAA similarity.

Therefore,
AC/BC = BC/DC
ACxDC =  (BC)² ......... (ii)

Now, add (i) and (ii)
ACxAD + ACxDC = (AB)² + (BC)²
AC(AD + DC) = (AB)² + (BC)²

From figure it is clear that
AD + DC = AC
Therefore,
ACxAC = (AB)² + (BC)²
(AC)² = (AB)² + (BC)² Proved.

Conservation of Linear Momentum

Conservation of Linear Momentum

          Momentum

    The product of mass and velocity of a moving body is said to be momentum. 

It is generally denoted by P. 

The SI unit of momentum is kilogram metre per second or kgms⁻¹. 

The dimension of momentum is [MLT⁻¹]

          Law of conservation of Linear Momentum

"The vector sum of the linear momenta of all the particles in an isolated system remains constant in the absence of any external force"

          Proof:

    Consider about an isolated system having n particles. Let the masses of the particles be M1, M2, M3 ..... Mn and their velocities V1, V2, V3 ....... Vn respectively. 

The vector sum of the linear momenta of all the particles in the system is given by,

P  = M1V1 + M2V2 + M3V3 + ....................... + MnVn .......................... (i)

Let the total mass of the system is M and velocity of the centre of mass of the system is Vcm

Form of Equation (i) will be

P = MVcm 

Now derivative both sides with respect to t

dP/dt = M(Vcm/dt)

dP/dt = Macm (acm is the acceleration of centre of mass of system)

dP/dt = Fext (Fext is external force applied on the system)

When Fext is Zero

dP/dt = 0

We know that derivative of a constant term is zero.

Therefore,

P = Constant

Hence,

P =  M1V1 + M2V2 + M3V3 + ....................... + MnVn  = Constant.