Question

The distance between Delhi and Agra is 200 km. A train travels the first 100 km at a speed of 50 km/h. How fast must the train travel the next 100 km, so as to average 70 km/h for the whole journey?

Answer 1

We have the following data to find the speed for the second part of the journey:
Total distance to be travelled by train (D) = 200 km
Average speed required (v_avg) = 70 km/hr
Time required for the entire journey (T):
$$\begin{gathered} \mathrm{Time}=\frac{\mathrm{Total}\mathrm{distance}}{\mathrm{Average}\mathrm{speed}} \\ =\frac{200}{70}\mathrm{hr} \\ T=\frac{20}{7}\mathrm{hr} \end{gathered}$$
For the first part of the trip:
Distance covered (d₁) = 100 km
Speed for this part of journey (v₁) = 50 km/hr
Time taken for the first part of journey:
$$\begin{gathered} \mathrm{Time}=\frac{\mathrm{Distance}\mathrm{travelled}}{\mathrm{Speed}} \\ \mathrm{So}, \\ {t}_1=\frac{100}{50}\mathrm{hr} \\ =2\mathrm{hr} \end{gathered}$$
For the second part of the trip,
Distance covered (d₂) = 100 km
Time taken for the second part of journey:
$$\begin{gathered} t_2=T-t_1 \\ =\left(\frac{20}{7}-2\right)\mathrm{hr} \\ =\frac{6}{7}\mathrm{hr} \end{gathered}$$
Speed of the train for the second part of the journey:
$$\begin{gathered} =\frac{\mathrm{distance}\mathrm{travelled}}{\mathrm{Time}} \\ =\frac{\left(100\right)\left(7\right)}{6} \\ =\boxed{116.67\mathrm{km}/\mathrm{hr}} \end{gathered}$$