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

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