Question

If a particle travels distance $s_1$, $s_2$ and $s_3$ taking equal time to cover the respective distances. Find average speed of the particle, if it travels with a velocity $v_1$, $v_2$ and $v_3$ for the respective distances.

Answer 1

Average speed of the particle = $\dfrac{v_1 + v_2 + v_3}{3}$

Answer 2

The problem states that a particle travels distances $s_1$, $s_2$, and $s_3$ taking equal time to cover each respective distance. Let this equal time be $t$. The velocities for these respective distances are $v_1$, $v_2$, and $v_3$.

We use the fundamental relationship between distance, velocity, and time:

$\text{Distance} = \text{Velocity} \times \text{Time}$

For each segment of the journey:

1. For distance $s_1$: $s_1 = v_1 t$
2. For distance $s_2$: $s_2 = v_2 t$
3. For distance $s_3$: $s_3 = v_3 t$

The average speed of the particle is defined as the total distance traveled divided by the total time taken.

$\text{Average Speed} (V_{avg}) = \dfrac{\text{Total Distance}}{\text{Total Time}}$

Calculate the Total Distance:

$\text{Total Distance} = s_1 + s_2 + s_3$

Substitute the expressions for $s_1$, $s_2$, and $s_3$:

$\text{Total Distance} = v_1 t + v_2 t + v_3 t$

Factor out $t$:

$\text{Total Distance} = (v_1 + v_2 + v_3)t$

Calculate the Total Time:

$\text{Total Time} = \text{Time for } s_1 + \text{Time for } s_2 + \text{Time for } s_3$

$\text{Total Time} = t + t + t$

$\text{Total Time} = 3t$

Now, substitute the Total Distance and Total Time into the average speed formula:

$V_{avg} = \dfrac{(v_1 + v_2 + v_3)t}{3t}$

The term $t$ cancels out from the numerator and the denominator:

$V_{avg} = \dfrac{v_1 + v_2 + v_3}{3}$

This result indicates that when a particle travels for equal time intervals with different velocities, its average speed is the arithmetic mean of those velocities.