Question

a person travels along a straight road for first t/3 time with speed v1 and next 2t/3 time with speed v2 what is the average speed

Answer 1

Answer:

$\longmapsto \rm { \dfrac{v_1 + v_2 2}{3} }$

Explanation:

As per the provided information in the given question, we have :

• A person travels along a straight road for first $\rm \dfrac{t}{3}$ time with speed $\rm v_1$.
• And, next $\rm \dfrac{2t}{3}$ time with speed of $\rm v_2$.

We are asked to calculate the average speed.

In order to calculate the average speed,we'll be using the formula of average speed.

$\longmapsto \boxed{ \bf {Speed_{(avg)} = \dfrac{Total\; distance}{Total \; time}}}$

We've to calculate the total distance and total time first.

Calculating total distance :

First case ::

• Speed = $\rm v_1$
• Time = $\rm \dfrac{t}{3}$

$\longmapsto \rm { Distance = Speed \times Time }$

$\longmapsto \rm { s_1 = v_1 \times \dfrac{t}{3} }$

$\longmapsto \boxed{\rm { s_1 = \dfrac{v_1 t}{3} }}$

Second case :

• Speed = $\rm v_2$
• Time = $\rm \dfrac{2t}{3}$

$\longmapsto \rm { Distance = Speed \times Time }$

$\longmapsto \rm { s_2 = v_2 \times \dfrac{2t}{3} }$

$\longmapsto \boxed{\rm { s_2 = \dfrac{v_22t}{3} }}$

Total distance ::

$\longmapsto \rm { Distance_{(Total)} = s_1 + s_2 }$

$\longmapsto \rm { Distance_{(Total)} =\dfrac{v_1 t}{3} + \dfrac{v_22t}{3} }$

$\longmapsto \rm { Distance_{(Total)} =\dfrac{v_1 t + v_2 2t}{3} }$

$\longmapsto \boxed{\bf { Distance_{(Total)} =\dfrac{t(v_1 + v_2 2)}{3} }}$

__________________________

Finding total time :

$\longmapsto \rm { Time_{(Total)} = Time_{(1st \; case)} + Time_{(Second \; case)} }$

$\longmapsto \rm { Time_{(Total)} =\dfrac{t}{3} + \dfrac{2t}{3} }$

$\longmapsto \rm { Time_{(Total)} =\dfrac{t + 2t}{3} }$

$\longmapsto \rm { Time_{(Total)} =\dfrac{3t}{3} }$

$\longmapsto \boxed{\bf { Time_{(Total)} = t }}$

__________________________

Calculating average speed :

$\longmapsto \rm {Speed_{(avg)} = \dfrac{Total\; distance}{Total \; time} }$

$\longmapsto \rm {Speed_{(avg)} = \dfrac{v_1 t + v_2 2t}{3} \div t }$

$\longmapsto \rm {Speed_{(avg)} = \dfrac{\not{t}(v_1 t + v_2 2)}{3} \times \dfrac{1}{\not{t}} }$

$\longmapsto \boxed{\bf {Speed_{(avg)} = \dfrac{v_1 + v_2 2}{3} }}$

∴ We got the required answer!!