Question

On a 100km road, a car travels the first 50km at a uniform speed of \[30kmh{{r}^{-1}}\]. Assuming constant velocity in the second half also, how fast must the car travel for the next 50km so as to have an average speed of \[45kmh{{r}^{-1}}\]for the entire journey?
\[\begin{align}
  & \text{A) 90kmh}{{\text{r}}^{\text{-1}}} \\
 & \text{B) 60kmh}{{\text{r}}^{-1}} \\
 & \text{C) 120kmh}{{\text{r}}^{-1}} \\
 & \text{D) 45kmh}{{\text{r}}^{-1}} \\
\end{align}\]

Answer 1

Hint: We know that the average speed is a measure to understand the time taken to reach destination taking into the odds of being slow or fast. The average speed can be easily calculated as we calculate the average for any given quantity.

Complete step by step solution:
The average speed of a body is defined as the speed required by the object to travel a distance in a particular time. The average speed is a quantity that gives a rough picture of the speed during the travel. The average speed can vary highly from the original value of speeds attained by the object during its course of travel.
The average speed of the car can be given as –
\[\text{Average speed}=\dfrac{\text{Total distance travelled}}{\text{Total time taken}}\]
We can find the time taken for each half of the travel using the speed and the distance given as –
\[\begin{align}
  & t=\dfrac{\text{Distance}}{\text{Speed}} \\
 & \Rightarrow {{t}_{1}}=\dfrac{50km}{30kmh{{r}^{-1}}} \\
 & \Rightarrow {{t}_{1}}=\dfrac{5}{3}hr \\
\end{align}\]
\[\begin{align}
  & {{t}_{2}}=\dfrac{50km}{x\text{ kmh}{{\text{r}}^{\text{-1}}}} \\
 & \Rightarrow {{t}_{1}}=\dfrac{50}{x}hr \\
\end{align}\]
The average speed of the car is given as 45 \[kmh{{r}^{-1}}\]. The speed with which the car have to travel in order to reach the 45\[kmh{{r}^{-1}}\]can be solved from the formula for the average speed of the car as –
\[\begin{align}
  & \text{Average speed}=\dfrac{\text{Total distance travelled}}{\text{Total time taken}} \\
 & \Rightarrow 45kmh{{r}^{-1}}=\dfrac{100}{\dfrac{5}{3}+\dfrac{50}{x}} \\
 & \Rightarrow 45(\dfrac{5}{3}+\dfrac{50}{x})=100 \\
 & \Rightarrow 15\left( 5x+150 \right)=100x \\
 & \Rightarrow 75x+2250=100x \\
 & \Rightarrow 25x=2250 \\
 & \therefore x=90kmh{{r}^{-1}} \\
\end{align}\]
The average speed with which the car has to move for the second half of the journey is \[90kmh{{r}^{-1}}\].

The correct answer is given by the option A.

Note:
We should be careful in calculating the average speed of the object as there is a chance of making a mistake by multiplying the given speed and the distance travelled to find the average. Always keep track with the dimensional formula to avoid mistakes.