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 find the average speed of the particle, we need to consider the entire journey and the total time taken. Here's the step-by-step solution:

1. Assume the total distance covered by the particle is \(2d\). Thus, each half is \(d\).
2. The particle covers the first half of the distance (\(d\)) at a speed of 6 m/s.
   • Time taken for the first half, \(t_1 = \frac{d}{6}\)
3. The second half of the distance (\(d\)) is covered in two equal time intervals with speeds of 9 m/s and 15 m/s.
   • Let the time for each of these intervals be \(t_2\).
   • Distance covered in the first interval: \(9 \times t_2\)
   • Distance covered in the second interval: \(15 \times t_2\)
   • Total distance for the second half: \(9 \times t_2 + 15 \times t_2 = d\)
   • Thus, \(24t_2 = d \Rightarrow t_2 = \frac{d}{24}\)
4. Total time taken for the journey:
   • Total time, \(t = t_1 + 2t_2 = \frac{d}{6} + 2 \times \frac{d}{24}\)
   • Simplifying, \(t = \frac{d}{6} + \frac{d}{12} = \frac{2d + d}{12} = \frac{3d}{12} = \frac{d}{4}\)
5. Average speed of the particle:
   • Average speed, \(v_{\text{avg}} = \frac{\text{total distance}}{\text{total time}} = \frac{2d}{\frac{d}{4}}\)
   • Simplifying, \(v_{\text{avg}} = 2d \times \frac{4}{d} = 8 \text{ m/s}\)

Thus, the average speed of the particle is 8 m/s, making the correct answer \(8 \text{ m/s}\).

Answer 2

Step 1: Analyze the motion

• Let the total distance covered by the particle be \(2S\).
• For the first half of the distance (\(S\)):
• For the second half of the distance (\(S\)), it is covered in two equal time intervals (\(t, t\)):
  • In the first interval (\(t\)), speed = \(9 \, \text{m/s}\), so: \[ S_1 = 9t \implies t = \frac{S_1}{9}. \]
  • In the second interval (\(t\)), speed = \(15 \, \text{m/s}\), so: \[ S_2 = 15t. \]
• Since \(S_1 + S_2 = S\), solve for \(t\): \[ 9t + 15t = S \implies t = \frac{S}{24}. \]

Step 2: Total time taken

The total time is:

\[ \text{Total time} = t_1 + 2t = \frac{S}{6} + 2 \cdot \frac{S}{24}. \]

Simplify:

\[ \text{Total time} = \frac{S}{6} + \frac{S}{12} = \frac{2S}{12} + \frac{S}{12} = \frac{3S}{12} = \frac{S}{4}. \]

Step 3: Average speed

The average speed is given by:

\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}. \]

Simplify:

\[ \text{Average speed} = \frac{2S}{\frac{S}{4}}. \]

\[ \text{Average speed} = \frac{2S \cdot 4}{S} = 8 \, \text{m/s}. \]

Final Answer: \(8 \, \text{m/s}\).