Question

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

• A. 8.8 m/s
• B. 10 m/s
• C. 9.2 m/s
• D. 8 m/s

Answer 1

The Correct Option is D

To determine the average speed of the particle, the total journey and its duration must be analyzed. The calculation proceeds as follows:

1. Let the total distance traveled by the particle be represented as \(2d\). Consequently, each half of the journey is of distance \(d\).
2. The initial \(d\) distance is covered at a speed of 6 m/s.
   • The time taken for this first segment is \(t_1 = \frac{d}{6}\).
3. The subsequent \(d\) distance is traversed in two equal time intervals, with speeds of 9 m/s and 15 m/s, respectively.
   • Assume the duration of each of these intervals is \(t_2\).
   • The distance covered during the first interval is \(9 \times t_2\).
   • The distance covered during the second interval is \(15 \times t_2\).
   • The sum of these distances constitutes the second half: \(9 \times t_2 + 15 \times t_2 = d\).
   • This simplifies to \(24t_2 = d\), yielding \(t_2 = \frac{d}{24}\).
4. The total time for the complete journey is calculated as follows:
   • Total time, \(t = t_1 + 2t_2 = \frac{d}{6} + 2 \times \frac{d}{24}\).
   • Upon simplification, \(t = \frac{d}{6} + \frac{d}{12} = \frac{2d + d}{12} = \frac{3d}{12} = \frac{d}{4}\).
5. The average speed of the particle is computed using the formula:
   • Average speed, \(v_{\text{avg}} = \frac{\text{total distance}}{\text{total time}} = \frac{2d}{\frac{d}{4}}\).
   • Simplifying this expression, \(v_{\text{avg}} = 2d \times \frac{4}{d} = 8 \text{ m/s}\).

Therefore, the particle's average speed is 8 m/s.