Question

A body covers half the distance with a speed of 20 m/s and the other half with a speed of 30 m/s. The average velocity of the body during the whole journey is

• A. zero
• B. 24 m/s
• C. 25 m/s
• D. None of these

Answer 1

The correct option is B

24 m/s

Step 1: Given data:

1. The speed of a body for the first half is 20 m/s.
2. The speed of a body for the other half is 30 m/s.

We know that;

1. Average velocity is the ratio of total displacement covered and total time taken by the body.
2. The formula for calculating the average velocity is $\frac{\mathrm{Total}\mathrm{displacement}}{\mathrm{Total}\mathrm{Time}\mathrm{taken}}$.
3. The formula for calculating the speed is $\mathrm{Speed}=\frac{\mathrm{Distance}}{\mathrm{Time}}$.

Now,

Let the total distance which the body covers be ‘d’.

And the time to cover the first half distance and second half distance be t₁ and t₂ respectively.

Step 2: Finding time taken to cover the first half distance:

In the first half, the body covers half the distance which means $\frac{d}{2}$.

The time taken to cover the first half distance is given by;

$\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{speed}}=\frac{{\displaystyle \left(\frac{d}{2}\right)}}{20}=\frac{d}{40}=t_1$

Step 3: Finding time taken to cover the second half distance:

The time taken to cover the second half distance is given by;

$=\frac{{\displaystyle \left(\frac{d}{2}\right)}}{30}=\frac{d}{60}=t_2$

Step 4: Calculating total time:

The formula for average velocity is $\frac{\mathrm{Total}\mathrm{displacement}}{\mathrm{Total}\mathrm{Time}\mathrm{taken}}$.

$$\begin{gathered} \mathrm{Total}\mathrm{Time}\mathrm{taken}=t_1+t_2=T \\ {t}_1+t_2=\frac{d}{40}+\frac{d}{60} \\ T=d\left(\frac{1}{40}+\frac{1}{60}\right) \\ =d\left(\frac{60+40}{60\times 40}\right) \\ =d\times \frac{100}{2400} \\ T=\frac{d}{24}. \end{gathered}$$

Step 5: Calculating average velocity:

The average velocity of the body;

$V_{avg}=\frac{d}{\left({\displaystyle \frac{d}{24}}\right)}=24m/s$

Therefore, the average velocity of the body during the whole journey is 24 m/s.

Hence, the correct option is (B).