Question

A particle moving along a straight line covers the first half of the distance with a speed of \( 3 \, \text{m/s} \), the other half of the distance is covered in two equal time intervals with speeds of \( 4.5 \, \text{m/s} \) and \( 7.5 \, \text{m/s} \) respectively, then the average speed of the particle during the motion is

• A. \( 4.0 \, \text{m/s}^{-1} \)
• B. \( 5.0 \, \text{m/s}^{-1} \)
• C. \( 5.5 \, \text{m/s}^{-1} \)
• D. \( 4.8 \, \text{m/s}^{-1} \)

Hint:

Average speed is defined as the total distance divided by the total time taken. It is crucial to distinguish between scenarios where distances are equal and scenarios where time intervals are equal, as the average speed calculation differs. If different distances are covered at different speeds, calculate the time for each segment and sum them up. If different speeds are maintained for equal time intervals, the average speed is simply the average of those speeds for that time interval. For complex scenarios, break the journey into segments and calculate total distance and total time.

Answer 1

The Correct Option is A

Step 1: Define variables for the total distance and time.
Let the total distance covered be \( 2D \).
The first half of the distance is \( D \).
The second half of the distance is also \( D \).
Step 2: Calculate the time taken for the first half of the distance.
Speed for the first half of the distance, \( v_1 = 3 \, \text{m/s} \).
Distance for the first half, \( d_1 = D \).
Time taken for the first half, \( t_1 = \frac{d_1}{v_1} = \frac{D}{3} \).
Step 3: Analyze the second half of the distance.
The second half of the distance (\( D \)) is covered in two equal time intervals.
Let each equal time interval be \( t_2 \).
The speeds for these intervals are \( v_2 = 4.5 \, \text{m/s} \) and \( v_3 = 7.5 \, \text{m/s} \) respectively.
Distance covered in the first part of the second half: \( d_{2a} = v_2 \cdot t_2 = 4.5 t_2 \).
Distance covered in the second part of the second half: \( d_{2b} = v_3 \cdot t_2 = 7.5 t_2 \).
The total distance for the second half is \( D = d_{2a} + d_{2b} \).
So, \( D = 4.5 t_2 + 7.5 t_2 = (4.5 + 7.5) t_2 = 12 t_2 \).
From this, we can find \( t_2 \) in terms of \( D \):
\( t_2 = \frac{D}{12} \).
The total time for the second half of the journey is \( T_2 = t_2 + t_2 = 2t_2 = 2 \cdot \frac{D}{12} = \frac{D}{6} \).
Step 4: Calculate the total distance and total time.
Total distance = \( 2D \).
Total time = \( t_1 + T_2 = \frac{D}{3} + \frac{D}{6} \).
To add these, find a common denominator (6):
Total time = \( \frac{2D}{6} + \frac{D}{6} = \frac{3D}{6} = \frac{D}{2} \).
Step 5: Calculate the average speed.
Average speed = \( \frac{\text{Total distance}}{\text{Total time}} \).
Average speed = \( \frac{2D}{\frac{D}{2}} = 2D \cdot \frac{2}{D} = 4 \, \text{m/s} \).