Question

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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 ,

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

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 ,

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

Now , Calculating Average speed of given body ;

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

Taking t common in numerator ,

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

• Cancelling t on both numerator and denominator ,

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

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

Answer 2

Case 1: When a Body covers distance lets say ‘D’ with speed ‘v1’ time taken can calculated by the following formula.

t1 = D/v1 , ……(1)

Case 2: When a Body covers distance lets say ‘D’ with speed ‘v2’ time taken can calculated by the following formula.

t2 = D/v2 , ……(2)

Case 3: When a Body covers distance lets say ‘D’ with speed ‘v3’ time taken can calculated by the following formula.

t3 = D/v3 , ……(3)

Adding up all the three statements we can get the total time taken by the body which comes out to be

t1 + t2 + t3 ;

or D/v1 + D/v2 + D/v3 ……..(4)

And the total distance covered by the body can be easily calculated as 3D.

So Applying the formula of Average velocity or speed,

Average velocity = Total distance covered/ Total Time Taken the body

Average velocity = 3D / (D/v1 + D/v2 + D/v3)

Solving out the equation results comes out to be:

(3 × v¹ * v² * v³) / (v¹v² + v²v³ + v³v¹)