A particle moving in a straight line covers half the distance with speed $6 \mathrm{~m} / \mathrm{s}$. The other half is covered in two equal time intervals with speeds $9 \mathrm{~m} / \mathrm{s}$ and $15 \mathrm{~m} / \mathrm{s}$ respectively. The average speed of the particle during the motion is :

<p>Let's denote the total distance covered by the particle as $2d$, where $d$ is the distance for each half. To calculate the average speed, we need to find the total distance traveled and divide it by the total time taken.</p> <p>For the first half of the journey, the particle covers the distance $d$ at a speed of $6 \, \text{m/s}$. The time taken for this part of the journey can be calculated using the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$. So,</p> $\text{time}_1 = \frac{d}{6}$ <p>For the second half of the journey, the distance $d$ is further divided into two parts, each covered in equal time intervals. Given the speeds are $9 \, \text{m/s}$ and $15 \, \text{m/s}$ respectively, let's call the equal time intervals $t$. The distances covered in these intervals can be found by $\text{distance} = \text{speed} \times \text{time}$.</p> <p>For the part covered at $9 \, \text{m/s}$:</p> $d_1 = 9t$ <p>For the part covered at $15 \, \text{m/s}$:</p> $d_2 = 15t$ <p>Since these two parts together make up the second half of the journey,</p> $d_1 + d_2 = d$ <br/><br/>$9t + 15t = d$ <p>This gives us $24t = d$, and from this, we can find $t = \frac{d}{24}$.</p> <p>The total time for the second half of the journey is the sum of the times for the two parts, which are equal ($t$ each), so the total time for the second half is $2t$. Since $t = \frac{d}{24}$,</p> $\text{time}_2 = 2 \times \frac{d}{24} = \frac{d}{12}$ <p>The total time taken for the entire journey is the sum of the times for the first and second halves:</p> $\text{total time} = \text{time}_1 + \text{time}_2 = \frac{d}{6} + \frac{d}{12}$ <br/><br/>$\text{total time} = \frac{2d}{12} + \frac{d}{12} = \frac{3d}{12} = \frac{d}{4}$ <p>The total distance is $2d$, and the total time is $\frac{d}{4}$. Therefore, the average speed is calculated as:</p> $$ \text{average speed} = \frac{\text{total distance}}{\text{total time}} = \frac{2d}{\frac{d}{4}} = \frac{2d}{1} \times \frac{4}{d} = 8 \, \text{m/s} $$ <p>Thus, the correct answer is <strong>Option D: 8 m/s</strong>.</p>