The answer of this should be V 2gy 12 And calculate the acc

The answer of this should be V = (2gy) ^1/2

And calculate the acceleration also
Use the \"EULERIAN DESCRIPTION METHOD\"

About a free falling ball with mass m and zero initial speed from a height y=0 by assuming no air drag on the ball. Please derive velocity field of this falling ball in Eulerian description method and calculate the acceleration of this falling ball from this Eulerian velocity field

Solution

solution:

here velocity field is function space and time coordinate,hence to calculate velocity field let use conservation of energy principle

at height y=0,ball posses potential energy due to height and as it moves by certain distance y in vertical direction it get converted into kinetic energy,hence

for ball

potential energy=kinetic energy

mgy=(1/2)*m*V^2

Vy=(2gy)^.5

as ball posses no velocity in x and z direction hence

vx=0

Vz=0

hence velocity vector is given by

V=Vxi+Vyj+Vzk

V=(2gy)^.5j

3) in eulerian method accelaration is function of space and time coordinate,hence as it has no velocity in x and z direction thus its ax=az=0

where ay=VxdVy/dx+VydVy/dy+VzdVy/dz+dvy/dt

as Vy is only function of y hence all other terms tends to zero

ay=Vy*dVy/dt=(2*g*y)^.5*dVy/dt

Vy=(2*g*y)^.5

dVy/dt=.5*(2*g)^.5*y^-(-.5)

on putting value we get

ay=g=9.81 m/s2

hence total accelaration of ball is given by

a=axi+ayj+azk

a=ayj

a=gj=9.81 j

hence magnitude is

a=(ay^2)^.5=ay=9.81 m/s2

5)velocity vector is given by

V=(2gy)^.5j

accelaration vector is given by

a=9.81 j