Question

A car travelled the total distance of $x$ in three equal intervals, first at a speed of $10\,kmh{r^{ - 1}}$. The second speed of $20\,kmh{r^{ - 1}}$ and the last third at a speed of $60\,kmh{r^{ - 1}}$. Determine the average speed of the car over the entire distance $x$.

Answer 1

Hint In this problem, the average speed of the car can be determined by using the speed or velocity formula with the relation of distance and time. In this question, the total distance is given, then the total time in terms of distance is determined, then the average speed of the car can be determined.

Formulae Used:
$v = \dfrac{d}{t}$
Where, $v$ is velocity or speed of the car, $d$ is the distance travelled by the car and $t$ is the time taken by the car.

Complete step-by-step solution:
Given that,
First speed of the car is, $10\,kmh{r^{ - 1}}$
First speed of the car in three intervals, ${v_1} = 30\,kmh{r^{ - 1}}$
Second speed of the car is, $20\,kmh{r^{ - 1}}$
Second speed of the car in three intervals, ${v_2} = 60\,kmh{r^{ - 1}}$
Third speed of the car is, $60\,kmh{r^{ - 1}}$
Third speed of the car in three intervals, ${v_3} = 180\,kmh{r^{ - 1}}$
The total distance, $d = x\,km$
Now,
Time taken, $t = \dfrac{d}{v}$, then
The total time taken,
$t = \dfrac{x}{{{v_1}}} + \dfrac{x}{{{v_2}}} + \dfrac{x}{{{v_3}}}\,.........\left( 1 \right)$
Now substituting the three speed values in the above equation (1), then the above equation is written as,
$t = \dfrac{x}{{30}} + \dfrac{x}{{60}} + \dfrac{x}{{180}}$
Now taking LCM in the above equation, then
$t = \dfrac{{60x + 30x + x}}{{180}}$
Now adding the terms in the numerator, then the above equation is written as,
$t = \dfrac{{10x}}{{180}}$
By cancelling the zeros in numerator and denominator, then
$t = \dfrac{x}{{18}}$
Thus, the above equation shows the total time taken.
Now, the average speed is
 $v = \dfrac{d}{t}\,..............\left( 2 \right)$
Substituting the total distance and the total time taken in the above equation (2), then
$v = \dfrac{x}{{\left( {\dfrac{x}{{18}}} \right)}}$
By arranging the terms in the above equation, then
$v = \dfrac{x}{1} \times \dfrac{{18}}{x}$
By cancelling the terms in the above equation, then
$v = 18\,kmh{r^{ - 1}}$
Thus, the above equation shows the average speed of the car.
$\therefore $ the average speed of the car is $18\,kmh{r^{ - 1}}$.

Note:- The given speed is multiplied by three because the car travels in three equal intervals. In equation (1) all the three distances are equal, because the car travels the same distance for three intervals, so all the three distances are equal to $x$.