Question

A body moves with velocity V= ln(x) where x is its position. The net force acting on the body is zero at x= ?​

Answer 1

Given:

Velocity of body= ln(x)

To Find:

The net force acting on the body is zero at x= 0

Solution:

We know that,

• Velocity v of a body is given by

$\pink{\boxed{\bf{v=\dfrac{ds}{dt}}}}$

where,

s is the displacement of the body at time t

• Acceleration a of a body is given by

$\purple{\boxed{\bf{a=\dfrac{dv}{dt}}}}$

where,

v is the velocity of the body at time t

• $\bf{\dfrac{d}{dx}(ln\:x)=\dfrac{1}{x}}$

• Force= mass × acceleration

$\rule{190}{1}$

Let the mass of body be 'm' and acceleration of body be 'a'

So,

$\longrightarrow\rm{a=\dfrac{dv}{dt}}$

On multiplying and dividing dx in R.H.S., we get

$\longrightarrow\rm{a=\dfrac{dv}{dt}\times\dfrac{dx}{dx}}$

$\longrightarrow\rm{a=\dfrac{dv}{dx}\times\dfrac{dx}{dt}}$

Here, $\rm{\dfrac{dx}{dt}}$ is the velocity of given body

So,

$\longrightarrow\rm{a=\red{\dfrac{d}{dx}(ln(x))}\times\orange{v}}$

$\longrightarrow\rm{a=\red{\dfrac{1}{x}}\times\orange{ln(x)}}$

$\longrightarrow\rm{a=\dfrac{ln(x)}{x}}$

$\rule{190}{1}$

Le the net force acting on given body be F

So,

$\longrightarrow\rm{F=ma}$

$\longrightarrow\rm{F=m\dfrac{ln(x)}{x}}$

It is also given that F= 0

So,

$\longrightarrow\rm{0=m\dfrac{ln(x)}{x}}$

$\longrightarrow\rm{m\dfrac{ln(x)}{x}=0}$

$\longrightarrow\rm{\dfrac{ln(x)}{x}=0}$

$\longrightarrow\rm{ln(x)=0}$

$\longrightarrow\rm{x=e^{0}}$

$\longrightarrow\rm\green{x=1}$

Hence, the net force acting on the body is zero at x= 1.

Answer 2

Answer:

X=1m

Explanation :

* V=lnx m/s (given)

* a=v dv/dx (lnx)

=> a = lnx d/dx (lnx)

* a=lnx/x

* Fnet = 0

=> a= 0

* lnx/x = 0

************** x = 1 ****************