Question

On a 120 km track , a train travels the first 30 km at a uniform speed of 30 km//h. How fast must the train travel the next 90 km so as to average 60 km//h for the entire trip?

Answer 1

$\huge{\underline{\boxed{\bf{\blue{Solution:-}}}}}$

$\bigstar{\underline{\sf{Given:-}}}$

• Distance d1 = 30 km

• Velocity v1 = 30 kmph

• Time taken = t1 = d1/v1 = 1 hour

$\huge{\underline{\boxed{\bf{\blue{Explaination:-}}}}}$

→ So distance remaining = d2 = 120 - d1 = 90 km

→ Let speed to cover distance d2 = v2  kmph

→ Time taken = d2/v2 = 90/v2  hours

→ Average speed = total distance / total duration of time

→ 60 kmph = 120 km / [1 hour + 90 / v2  hours.

→ 1 = 2 / [ 1 + 90 / v2 ]

→ $\boxed{\sf{v2 = 90 kmph}}$

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★ Alternative way:-

→ As the total distance = 120 km

→ Average speed = 60 kmph 

→Total time duration = 120 / 60 = 2 hours 

→ Time taken for the first 30 km =  30 km /30 kmph = 1 hours

→ Hence, time remaining to cover the remaining 90km = 2 hrs - 1 hrs = 1 hour

→ Speed during 90 km 

→ $\boxed{\sf{= 90 kmph}}$

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Answer 2

Answer:

$\large{\underline{\boxed{\mathfrak\green{Next \: speed \: should \: be = 90\: km/h }}}}$

Given:-

• Total distance (d) = 120 km
• 2nd distance (d2) = 90 km
• Time taken in 1st distance (t1) = 1 hour.
• Average speed = 60 km/h

To find:-

• Speed in 2nd distance = ?

Let speed in 2nd distance be $\sf v_2$

$\dashrightarrow\sf Time \: in \: 2nd \: distance =\dfrac{Distance_2}{Speed_2} \\\\\dashrightarrow\sf T_2 = \dfrac{90}{v_2}\dots \dots (eq.1)$

$\rm We \: know, \\\\\star \: {\boxed{\rm{Average \: speed =\dfrac{Total \: distance}{Total \: time} }}}$

$\rm \: \: [Putting \: values :- ]$

$\dashrightarrow\sf 60 = \dfrac{30 + 90}{1 + \dfrac{90}{v_2}} \: \: \: [From \: (eq.1)]$

$\dashrightarrow\sf 60 = \dfrac{120}{\dfrac{v_2 + 90}{v_2}}$

$\dashrightarrow\sf 60 = 120 \times \dfrac{v_2}{v_2 + 90}$

$\dashrightarrow\sf 60 =\dfrac{120v_{2}}{v_2 + 90}$

$\dashrightarrow\sf 60(v_2 + 90) = 120v_{2}$

$\dashrightarrow\sf 60v_{2} + 5400 = 120v_{2}$

$\dashrightarrow\sf 120v_{2} - 60v_{2} = 5400$

$\dashrightarrow\sf 60v_{2} = 5400$

$\dashrightarrow\sf v_2 = 5400/60$

$\dashrightarrow\large{\underline{\boxed{\sf\blue{v_2 = 90 \: km/h }}}}$

${\underline{\underline{\therefore{\rm{Speed \: in \: 2nd \: distance = 90 \: km/h }}}}}$