Question

A car travels at a speed of 80 km/h during the first half of its running time and at 40 km/h during the other half, then the average speed of the car …………
A) $50km/hr$
B) $75km/hr$
C) $60km/hr$
D) $40km/hr$

Answer 1

The average speed is the rate of the total traversed distance with respect to the total time spent. This is the measure of an overall speed from the different speeds that the object possesses at different times throughout the total interval of motion. Here, you need to understand that the journey is described in two equal time intervals.

Formulae Used:
The average speed ${V_{avg}}$ of an object can be defined as
${V_{avg}} = \dfrac{{{d_{total}}}}{{{T_{total}}}}$
where, ${d_{total}}$ is the total distance traversed throughout the journey and the ${T_{total}}$ is the total time interval of the journey.

Complete step by step answer:
Given:
The car has a speed of $80km/hr$ in the first half of its running time.
The car has a speed of $40km/hr$ in the second half of its running time.
To get: The average speed of the car.
Step 1:
In the first half of the running time the car runs with a speed ${v_1} = 80km/hr$
Let the total running time of the car is $T$ hr.
Hence, the car runs with the speed ${v_1}$ for a time interval $\dfrac{T}{2}hr$.
Calculate the distance ${d_1}$ traversed by the car in the first time interval
${d_1} = {v_1}\dfrac{T}{2} \\\ \Rightarrow {d_1} = 80 \times \dfrac{T}{2}km \\\ \Rightarrow {d_1} = 40Tkm \\\$
Step 2:
In the second half of the running time the car runs with a speed ${v_2} = 40km/hr$
Hence, the car runs with the speed ${v_2}$ for a time interval $\dfrac{T}{2}hr$.
Calculate the distance ${d_2}$ traversed by the car in the first time interval
${d_2} = {v_2}\dfrac{T}{2} \\\ \Rightarrow {d_2} = 40 \times \dfrac{T}{2}km \\\ \Rightarrow {d_2} = 20Tkm \\\$
Step 3:
Now, the average velocity of ${v_{avg}}$ is the rate of the total distance traveled with respect to the total time.
So, calculate the total distance
${d_{total}} = {d_1} + {d_2} \\\ \Rightarrow {d_{total}} = \left( {40T + 20T} \right)km \\\ \Rightarrow {d_{total}} = 60Tkm \\\$
The total time of journey is ${T_{lotal}} = Thr$
Calculate the average velocity ${v_{avg}}$ from the eq (1)
${v_{avg}} = \dfrac{{{d_{total}}}}{{{T_{total}}}} \\\ \Rightarrow {v_{avg}} = \dfrac{{60T}}{T}km/hr \\\ \therefore {v_{avg}} = 60km/hr \\\$

A car travels at a speed of 80 km/h during the first half of its running time and at 40 km/h during the other half, then the average speed of the car $60km/hr$. Hence, Option (C) is correct.

Note:
The average speed of ${v_{avg}}$ can be computed from an easier approach. The car runs with ${v_1}$ speed with half of the total time and with ${v_2}$ speed with the other half of the total time. Hence, you can calculate the average speed ${v_{avg}}$ as the average of the two velocities ${v_1}$ and ${v_2}$ .
${v_{avg}} = \dfrac{{{v_1} + {v_2}}}{2} \\\ \Rightarrow {v_{avg}} = \dfrac{{80 + 40}}{2}km/hr \\\ \Rightarrow {v_{avg}} = \dfrac{{120}}{2}km/hr \\\ \therefore {v_{avg}} = 60km/hr \\\$
Hence, you get the average velocity ${v_{avg}} = 60km/hr$.