Question

. Abody covers one-third of the distance with a velocity
v, the second one-third of the distance with a velocity
V2, and the last one-third of the distance with a velocity
Vg. The average velocity is :
in + V₂ + V₂
(1)
3
12
3v, v. V
2 + V₂ V g + Vgvi
vy, +V,V, +V, V,
cas y​

Answer 1

$\huge{\bold{\underline{\underline{Solution:-}}}}$

$\large{\bold{\underline{To Find }}}$

Average velocity of the particle.

I.e. ${v_ a = (to find)}$

$\large{\bold{\underline{\underline{Step - by - Step - Explanation}}}}$

Let the body covers a distance "S" in velocities ${v_1,v_2,v_3}$ respectively.

and the time covered in each distances be ${t_1,t_2,t_3}$ seconds.

${\therefore t_1 = \frac{S}{v_1}}$

${\therefore t_2 = \frac{S}{v_2}}$

${\therefore t_3 = \frac{S}{v_3}}$

Now,Average velocity formula

$\huge{\boxed{\boxed{Average velocity = \frac{Total Displacement}{Total time taken}}}}$

Substituting the values,

${v_a = \frac{S + S + S}{ \frac{S}{v_1} + \frac{{S}}{v_2} + \frac{S}{v_3}}}$

${v_a = \frac{3S}{ S \times (\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3})}}$

Displacement gets cancel on both numerator and denominator.

${v_a = \frac{3}{ (\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3})}}$

${v_a = \frac{3}{ \frac{v_2 v_3 + v_3 v_1 + v_1 v_2 }{v_1 v_2 v_3}}}$

${v_a = \frac{3v_1 v_2 v_3}{v_1v_2 + v_2v_3 + v_3v_1}}$

$\huge{\boxed{\boxed{v_a = \frac{3v_1 v_2 v_3}{v_1v_2 + v_2v_3 + v_3v_1}}}}$

Hence derived.

Answer 2

$\huge{\underline{\underline{\sf{Answer \colon}}}}$

Let v,v' and v" be the velocities of the body during the whole time interval

Let S be the distance travelled in each interval

We know that,

$\boxed{ \sf{v = \frac{displacement}{time} }}$

Thus,

• In time t,the object would travel S/v metres

• In time t',the object would travel S/v' metres

• In time t",the object would travel S/v" metres

We need to find the Average Velocity of the body

Implies,

$\boxed{\sf{average \: velocity = \frac{total \: displacement}{t.time \: taken} }}$

Thus,

Average Velocity = (S+S+S)/t + t' + t"

→ A.Velocity = 3S/(S/v + S/v' + S/v")

→A.Velocity = 3(1/v + 1/v' + 1/v")

→A.Velocity = 3vv'v"/(vv'+ v'v"+ v"v)