Question

A particle covers half of its total distance with speed $v_1$ and the rest half distance with speed $v_2$.Its average speed during the complete journey is :

• A. $\frac{v_1+v_2}{2}$
• B. $\frac{v_1{v}_2}{v_1+v_2}$
• C. $\frac{2v_1{v}_2}{v_1+v_2}$
• D. $\frac{v_1^2{v}_2^2}{v_1^2+v_2^2}$

Answer 1

The correct option is C $\frac{2v_1{v}_2}{v_1+v_2}$

If The half distance (x) covered with the speed $v_1$ in $t_1$ time.

Using formula of speed, $v_1=\frac{x}{t_1}$

so, $t_1=\frac{x}{v_1}$

And another half distance (x), covered with speed $v_2$ in time $t_2$.

so, $v_2=x{t}_2$

$t_2=\frac{x}{v_2}$

$\text{A}verage\text{V}elocity=\frac{Totaldistance}{Totaltime}$

Total time$=t_1+t_2=\frac{x}{v_1}+\frac{x}{v_2}$

$=\frac{(v_2\ast x+v_1\ast x)}{v_1{v}_2}$

Total distance $=x+x=2x$

On putting the values of total distance and total time in the formula of average speed, we get

Average speed$=\frac{2x}{\left(\frac{v_{2}x+v_{1}x}{v_1\ast v_2}\right)}$

$V=\frac{2v_1{v}_2}{(v_1+v_2)}$