Question

if a body travels with speed v (1),v (2),v (3) for equal intervals of time ,find the average speed . ​
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Answer 1

Given :

• A body travelled with speeds v₁ , v₂ and v₃ for equal intervals of time

To Find :

• The average speed of the body

Solution :

Let the time interval be "t"

Since ,

$\begin{gathered}\begin{gathered} \\ \star \: {\boxed{\purple{\sf{distance = speed \times time}}}} \\ \\ \end{gathered}\end{gathered}$

Distance covered with speed v₁ in time t is v₁t

Distance covered with speed v₂ in time t is v₂t

Distance covered with speed v₁ in time t is v₃t

Total Time taken in these three intervals is t + t + t = 3t [given condition]

Now , Average speed of a body is nothing but the ratio between total distance travelled and total time taken.If the Mathematically ,

$\begin{gathered}\begin{gathered} \\ \star \: {\boxed{\purple{\sf{ v_{avg} = \dfrac{total \: distance}{total \: time \: taken}}}}} \\ \\ \end{gathered}\end{gathered}$

Now , Calculating Average speed of given body ;

$\begin{gathered}\begin{gathered} \\ : \implies \sf v_{avg} \: = \dfrac{v_1t + v_2t + v_3t }{t + t + t} \\ \\ \end{gathered}\end{gathered}$

Taking t common in numerator ,

$\begin{gathered}\begin{gathered} \\ : \implies \sf \: v_{avg} = \dfrac{t(v_1 + v_2 + v_3)}{3t} \\ \\ \end{gathered}\end{gathered}$

Cancelling t on both numerator and denominator ,

$\begin{gathered}\begin{gathered} \\ : \implies{\underline{\boxed{\mathfrak{ \pink{ \: v_{avg} = \dfrac{v_1 + v_2 + v_3}{3} }}}}} \: \bigstar \\ \\ \end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered} \\ \therefore {\underline{\sf{Hence \: , \: The \: average \: speed \: of \: the \: given \: body \: is \: \bold{ \frac{v_1 + v_2 + v_3}{3} }}}} \\ \end{gathered}\end{gathered}$

Answer 2

Answer:

Given :

A body travelled with speeds v₁ , v₂ and v₃ for equal intervals of time

To Find :

The average speed of the body

Solution :

Let the time interval be "t"

Since ,

\begin{gathered}\begin{gathered}\begin{gathered} \\ \star \: {\boxed{\purple{\sf{distance = speed \times time}}}} \\ \\ \end{gathered}\end{gathered}\end{gathered}

⋆

distance=speed×time

Distance covered with speed v₁ in time t is v₁t

Distance covered with speed v₂ in time t is v₂t

Distance covered with speed v₁ in time t is v₃t

Total Time taken in these three intervals is t + t + t = 3t [given condition]

Now , Average speed of a body is nothing but the ratio between total distance travelled and total time taken.If the Mathematically ,

\begin{gathered}\begin{gathered}\begin{gathered} \\ \star \: {\boxed{\purple{\sf{ v_{avg} = \dfrac{total \: distance}{total \: time \: taken}}}}} \\ \\ \end{gathered}\end{gathered}\end{gathered}

⋆

v

avg

=

totaltimetaken

totaldistance

Now , Calculating Average speed of given body ;

\begin{gathered}\begin{gathered}\begin{gathered} \\ : \implies \sf v_{avg} \: = \dfrac{v_1t + v_2t + v_3t }{t + t + t} \\ \\ \end{gathered}\end{gathered}\end{gathered}

:⟹v

avg

=

t+t+t

v

1

t+v

2

t+v

3

t

Taking t common in numerator ,

\begin{gathered}\begin{gathered}\begin{gathered} \\ : \implies \sf \: v_{avg} = \dfrac{t(v_1 + v_2 + v_3)}{3t} \\ \\ \end{gathered}\end{gathered} \end{gathered}

:⟹v

avg

=

3t

t(v

1

+v

2

+v

3

)

Cancelling t on both numerator and denominator ,

\begin{gathered}\begin{gathered}\begin{gathered} \\ : \implies{\underline{\boxed{\mathfrak{ \pink{ \: v_{avg} = \dfrac{v_1 + v_2 + v_3}{3} }}}}} \: \bigstar \\ \\ \end{gathered}\end{gathered} \end{gathered}

:⟹

v

avg

=

3

v

1

+v

2

+v

3

★

\begin{gathered}\begin{gathered}\begin{gathered} \\ \therefore {\underline{\sf{Hence \: , \: The \: average \: speed \: of \: the \: given \: body \: is \: \bold{ \frac{v_1 + v_2 + v_3}{3} }}}} \\ \end{gathered}\end{gathered}\end{gathered}

∴

Hence,Theaveragespeedofthegivenbodyis

3

v

1

+v

2

+v

3