Question

the velocity of the particle of mass M moving along x-axis is given by v= b√x ,where b is a constant find work done by the force acting on the particle during its Motion from x = 0 to x = 4 metre​

Answer 1

Answer:

Given:

Mass of object = M

Velocity function wrt displacement is as follows :

v = b√x

To find:

Work done during the motion

Concept:

As per Work-Energy Theorem, we can say that the total work done by all the forces will be equal to change in Kinetic Energy in that reference frame.

$\boxed{ \red{ \huge{work \: done \: = \Delta KE}}}$

Calculation:

$v = b \sqrt{x}$

So Kinetic energy function wrt to displacement :

$KE = \frac{1}{2} M {v}^{2}$

$= > KE = \frac{1}{2} M {(b \sqrt{x} )}^{2}$

$= > KE = \frac{1}{2} M {b}^{2} x$

Now Kinetic energy at x = 0 :

$KE1= \frac{1}{2} M {b}^{2} \times 0 = 0$

Now Kinetic energy at x = 4 m

$KE2= \frac{1}{2} M {b}^{2} \times 4 = 2M {b}^{2}$

So work done is change in KE :

$work \: done \: = \Delta KE$

$= > work \: done \: =2M {b}^{2} - 0$

$= > work \: done \: =2M {b}^{2}$

So final answer :

$\boxed{ \blue{ \huge{ \bold{work \: done \: =2M {b}^{2} }}}}$

Answer 2

$\large{\underline{\underline{\mathfrak{Answer :}}}}$

• Work done is 2mb²

$\rule{200}{0.5}$

$\underline{\underline{\mathfrak{Step-By-Step-Explanation :}}}$

Given :

• velocity (v) = b√x

________________________

To Find :

• Work done by force between x = 0 to x = 4

__________________________

Solution :

✯ Initial velocity (u) :

• x = 0
• u = b√0 = 0 m/s

________________

✯ Final velocity (v)

• x = 4
• v = b√4 m/s

______________________________

As we know that :

$\large{\boxed{\sf{\Delta K.E \: = \: W}}} \\ \\ \implies {\sf{\Delta K.E \: = \: \dfrac{1}{2} mv^2 \: - \: \dfrac{1}{2} mu^2}} \\ \\ \implies {\sf{\Delta K.E \: = \: \dfrac{1}{2} m(v^2 \: - \: u^2)}} \\ \\ \implies {\sf{\Delta K.E \: = \: \dfrac{1}{2} m \big[ (b \sqrt{4})^2 \: - \: 0^2 \big]}} \\ \\ \implies {\sf{\Delta K.E \: = \: \dfrac{1}{2} m(b^2 4)}} \\ \\ \implies {\sf{\Delta K.E \: = \: 2mb^2}} \\ \\ \implies {\sf{W \: = \: 2mb^2}}$