Question

If a body covers a distance x with speed v₁ and then it covers another distance x with speed v₂, find the average speed for the whole journey.

Answer 1

Step 1: Definition of speed, average speed

1. The rate of change of distance with respect to time is called speed. So speed = distance ÷time. From this, the formula for time will be

time =$\frac{dis\tan ce}{velocity}$.

2. average speed of a body can be calculated by dividing the total distance traveled by the body by the total time taken by the body.

So average speed =$\frac{totaldis\tan ce}{totaltime}$.

Step 2: Calculating average speed

1. Let t₁ be the time taken by the body to cover a distance x with speed v₁
   
2. t₂ is the time taken by the body to cover another distance x with speed v₂.
3. So $t_1=\frac{x}{v_1}$ and$t_2=\frac{x}{v_2}$.
4. Now average speed = $\frac{totaldis\tan ce}{totaltime}$= $\frac{x+x}{t_1+t_2}$= $\frac{2x}{{\displaystyle \frac{x}{v_1}}+{\displaystyle \frac{x}{v_2}}}$= $\frac{2x}{x({\displaystyle \frac{1}{v_1}}+{\displaystyle \frac{1}{v_2}})}$=$\frac{2}{\left({\displaystyle \frac{1}{v_1}}+{\displaystyle \frac{1}{v_2}}\right)}$= $\frac{2v_1{v}_2}{v_1+v_2}$.
   Final Answer: average speed =$\frac{2v_1{v}_2}{v_1+v_2}$.