Question

A car travels along a straight line for the first half time with the speed of $40\dfrac{km}{h}$ and the second half time with the speed of $50\dfrac{km}{h}$. Find out the mean speed of the car.

Answer 1

Hint: The average speed or the mean speed of the particle can be found out by taking the ratio of the total distance travelled by the total time taken for the travel. The velocities are given and the time taken are given. Using this the distance can be calculated as the product of speed and the time taken. By this method calculate the distance travelled at each speed and find the mean speed of the particle. These all may help you to solve this question.

Complete answer:
It has been mentioned that the body is moving for the first half period of time with a speed which is given as,
${{v}_{1}}=40\dfrac{km}{h}$
The time taken for this will be given as,
${{t}_{1}}=\dfrac{t}{2}$
Similarly for the second half period of time, the speed of the motion is given as,
${{v}_{2}}=50\dfrac{km}{h}$
And also the time period will be,
${{t}_{2}}=\dfrac{t}{2}$
Therefore the distance travelled in the first half time will be given as,
${{x}_{1}}={{v}_{1}}\times {{t}_{1}}$
Substituting the values in it,
${{x}_{1}}=40\times \dfrac{t}{2}$
And the distance travelled in the second half time will be given as,
${{x}_{2}}={{v}_{2}}\times {{t}_{2}}$
\[{{x}_{2}}=50\times \dfrac{t}{2}\]
The total distance travelled can be calculated now. That is,
\[d={{x}_{1}}+{{x}_{2}}=40\times \dfrac{t}{2}+50\times \dfrac{t}{2}=45t\]
The total time taken will be,
\[t={{t}_{1}}+{{t}_{2}}=\dfrac{t}{2}+\dfrac{t}{2}=t\]
Therefore the mean speed will be given as,
\[s=\dfrac{\text{total distance}}{\text{total time}}=\dfrac{d}{t}\]
Substituting the values in it will give,
\[s=\dfrac{45t}{t}=45\dfrac{km}{h}\]

Note:
Speed is given as a scalar quantity. It is given as the rate at which the body covers distance. The average speed is the distance which is also a scalar quantity per time ratio. The average velocity is defined as the displacement or position change which is a vector quantity per time ratio.