A particle travelling in one direction on a straight road covers half the distance with a velocity 30 m/s. The remaining part of the distance is covered with velocity 10 m/s for half the time and with velocity 30 m/s for the other half of the time. Find the magnitude of the average velocity of the particle over the whole time of motion

To solve this problem, we need to find the total distance covered and the total time taken, and then divide the total distance by the total time to find the average velocity.

Step 1: Find the total distance covered Let's assume the total distance covered by the particle is D. According to the problem, half of this distance is covered at a velocity of 30 m/s, and the other half is covered at velocities of 10 m/s and 30 m/s for equal amounts of time. So, the total distance D is divided into two equal parts, D/2 and D/2.

Step 2: Find the total time taken The time taken to cover the first half of the distance (D/2) at a velocity of 30 m/s can be found using the formula time = distance/velocity. So, the time taken for the first half is (D/2) / 30. For the second half of the distance (D/2), half of the time it's covered at 10 m/s and the other half at 30 m/s. So, the time taken for the second half is (D/4) / 10 + (D/4) / 30, which simplifies to 7D/120. So, the total time taken is (D/2) / 30 + 7D/120.

Step 3: Find the average velocity The average velocity is the total distance divided by the total time. So, the average velocity is D / [(D/2) / 30 + 7D/120].

Step 4: Simplify the expression If we simplify this expression, we find that the average velocity is approximately 20 m/s.