Question

A car travels at a velocity of 80 km/h during the first half of its running time and at 40km/h during the other half. Find the average speed of the car.

Answer 1

Hint: Average velocity of a body is defined as the total distance cover by a body divided by the total time taken by the body to cover this distance. Find the distances travelled by the car in both the half times by using $\text{distance = speed }\!\!\times\!\!\text{ time}$. Then add the distances and divide by the total time.

Formula used:
$\text{distance = speed }\!\!\times\!\!\text{ time}$
${{v}_{av}}=\dfrac{d}{t}$

Complete step-by-step solution:
It is given that a car travels for a certain distance. It covers this distance for two different constant velocities. It is said that the car travels at a constant velocity of 80 km/h during the first half of the total time it takes to cover the distance. After the half time, the car instantaneously decreases its velocity to 40km/h and covers the remaining distance with this constant speed.
Now we have to calculate the average velocity of the car for this time.
Let us first understand what is meant by the average velocity of a body.
The average velocity of a body is defined as the total distance covered by a body divided by the total time taken by the body to cover this distance.
If the total distance covered is d and the total time taken is t, then
${{v}_{av}}=\dfrac{d}{t}$ …. (i)
Therefore, we have to calculate the total distance traveled by car.
Let the total time taken by the car be t. This means that the time interval in both the halves of the journey is $\dfrac{t}{2}$.
To calculate the distance in both halves, we will use the formula $\text{distance = speed }\!\!\times\!\!\text{ time}$.
Hence, the distance travelled in the first half is ${{d}_{1}}=80\left( \dfrac{t}{2} \right)=40t$.
And the distance travelled in the second half is ${{d}_{2}}=40\left( \dfrac{t}{2} \right)=20t$.
Since, $d={{d}_{1}}+{{d}_{2}}$.
$\Rightarrow d=40t+20t=60t$.
Substitute the value of d in equation (i).
${{v}_{av}}=\dfrac{60t}{t}=60km/h$.
This means that the average speed of the car is 60km/h.

Note: If a body travels with a constant speed ${{v}_{1}}$ for the first half of the total time for its journey and travels the rest distance with a constant speed of ${{v}_{2}}$, then its average speed is given as ${{v}_{av}}=\dfrac{{{v}_{1}}+{{v}_{2}}}{2}$.
We can also use this formula to find the average velocity of the car.