A car travels from Kota to Jaipur with speed 30 km/h, and it returns along the same path with speed 60 km/h. Calculate average speed of the car and Avg. velocity?

Step-by-step solution

1. 1

   Given speeds

   Speed from Kota to Jaipur $v_1 = 30 \text{ km/h}$

   $$v_1 = 30 \text{ km/h}$$
2. 2

   Given speeds

   Speed from Jaipur to Kota $v_2 = 60 \text{ km/h}$

   $$v_2 = 60 \text{ km/h}$$
3. 3

   Definitions

   Average speed = $\frac{\text{Total Distance}}{\text{Total Time}}$

   $$\text{Avg. Speed} = \frac{D_{total}}{T_{total}}$$
4. 4

   Definitions

   Average velocity = $\frac{\text{Total Displacement}}{\text{Total Time}}$

   $$\vec{v}_{avg} = \frac{\Delta \vec{x}}{T_{total}}$$
5. 5

   Assume distance

   Let distance between Kota and Jaipur be $D$

   $$D$$
6. 6

   Calculate total displacement

   Car returns to starting point, so total displacement is $0$

   $$\Delta \vec{x} = 0$$
7. 7

   Calculate average velocity

   Average velocity = $\frac{0}{T_{total}} = 0 \text{ km/h}$

   $$\vec{v}_{avg} = 0 \text{ km/h}$$
8. 8

   Calculate time for each leg

   Time for first leg $t_1 = \frac{D}{v_1} = \frac{D}{30}$

   $$t_1 = \frac{D}{30}$$
9. 9

   Calculate time for each leg

   Time for second leg $t_2 = \frac{D}{v_2} = \frac{D}{60}$

   $$t_2 = \frac{D}{60}$$
10. 10

    Calculate total distance

    Total distance $D_{total} = D + D = 2D$

    $$D_{total} = 2D$$