Problem Overview

We need to find the speed '$v$' (in km/hr) for the second part of a car's journey. The journey is split into two parts based on distance, with different speeds and a known overall average speed.

• Part 1: Distance = $\frac{1}{4}$ Total Distance, Speed = 60 km/hr
• Part 2: Distance = $\frac{3}{4}$ Total Distance, Speed = $v$ km/hr
• Average Speed (Whole Journey) = 50 km/hr

Calculating Time Components

Let the total distance be $D$. Time is calculated using the formula: Time = Distance / Speed.

• Time for Part 1 ($t_1$): Calculated as $\frac{\text{Distance}_1}{\text{Speed}_1} = \frac{D/4}{60} = \frac{D}{240}$ hr
• Time for Part 2 ($t_2$): Calculated as $\frac{\text{Distance}_2}{\text{Speed}_2} = \frac{3D/4}{v} = \frac{3D}{4v}$ hr
• Total Time ($T$): The sum of the times for both parts, $T = t_1 + t_2 = \frac{D}{240} + \frac{3D}{4v}$ hr

Applying Average Speed Formula

The formula for average speed is: Average Speed = Total Distance / Total Time.

Substitute the known values into the formula:

$50 = \frac{D}{\frac{D}{240} + \frac{3D}{4v}}$

Simplify the equation by dividing the numerator and denominator by $D$ (since $D \neq 0$):

$50 = \frac{1}{\frac{1}{240} + \frac{3}{4v}}$

To solve for $v$, first take the reciprocal of both sides:

$\frac{1}{50} = \frac{1}{240} + \frac{3}{4v}$

Solving for Unknown Speed v

Isolate the term containing $v$ on one side of the equation:

$\frac{3}{4v} = \frac{1}{50} - \frac{1}{240}$

To subtract the fractions, find a common denominator, which is 1200:

$\frac{3}{4v} = \frac{24}{1200} - \frac{5}{1200}$

$\frac{3}{4v} = \frac{19}{1200}$

Now, solve for $v$ by cross-multiplying:

$4v \times 19 = 3 \times 1200$

$76v = 3600$

Finally, calculate the value of $v$:

$v = \frac{3600}{76}$

Simplify the fraction by dividing the numerator and denominator by 4:

$v = \frac{900}{19}$

The calculated speed $v$ is $\frac{900}{19}$ km/hr.