`int(dv)/(vsqrt(1-v^2))=`

Evaluate: (i) int1/(sqrt(a^2-b^2\ x^2))\ dx (ii) int1/(sqrt((2-x)^2+1))\ dx

Let I_(1)=int_(1)^(2)(x)/(sqrt(1+x^(2)))dx and I_(2)=int_(1)^(2)(1)/(x)dx .Then

The Speed Of a particle moving in a plane is equal to the magnitude of its instantaneous velocity, v=|vl= sqrt(v_x^2+v_y^2). (a) Show that the rate of change of the speed is (dv)/(dt)=(v_xa_x+v_ya_y)/(sqrt(v_x^2+v_y^2)) . (b) Show that the rate of change of speed can be expressed as (dv)/(dt) is equal to a_t the component of a that is parallelto v.

A block is mass m moves on as horizontal circle against the wall of a cylindrical room of radius R. The floor of the room on which the block moves is smooth but the friction coefficient between the wall and the block is mu . The block is given an initial speed v_0 . As a function of the speed v write a. the normal force by the wall on the block. b. the frictional force by the wall and c. the tangential acceleration of the block. d. Integrate the tangential acceleration ((dv)/(dt)=v(dv)/(ds)) to obtain the speed of the block after one revoluton.

If p=9, v=3 , then find p is terms of v from the equation (dp)/(dv)=v+1/v^2

Statement I: The work done on an ideal gas in changing its volume from V_(1) to V_(2) under a polytropic porcess is given by the integral int _(v_(1))^(v_(2)) P. Dv taken along the process Statement II: No work is done under an isochroci process of the gas.

If v=(t+2)(t+3) then acceleration (i.e(dv)/(dt)) at t=1 sec.

Show that, v = A/r + B satisfies the differential equation (d^2v)/(dr^2)+2/r.(dv)/(dr)=0

The motion of a body falling from rest in a medium is given by equation, (dv)/(dt)=1-2v . The velocity at any time t is

int_(1)^(2)((x^(2)-1)dx)/(x^(3).sqrt(2x^(4)-2x^(2)+1))=(u)/(v) where u and v are in their lowest form. Find the value of ((1000)u)/(v) .