Question

A car covers the first half the distance between two places at $40km{h^{ - 1}}$ another half at $60km{h^{ - 1}}$, the average speed of car is:

Answer 1

Hint: For this we have to use the formula of velocity that is the ratio of displacement and time. After that calculate time for both conditions because average velocity is the ratio of total displacement and total time.

Complete step by step solution:
First of all, The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to the object's speed and direction of motion. Therefore it is a vector quantity.

Let us consider, $2x$ is total distance covered by the car,
Time taken to complete the half distance that is $x$ with velocity $v$ equal to $40km{h^{ - 1}}$
$\therefore v = \dfrac{x}{{{t_1}}}$
Insert value of v in the equation
$ \Rightarrow {t_1} = \dfrac{x}{{40}}hr$
And the time taken to complete another half distance $x$ with velocity $v$ equal to $60km{h^{ - 1}}$
Again by definition of velocity
$v = \dfrac{x}{{{t_2}}}$
Insert value of velocity
$ \Rightarrow {t_2} = \dfrac{x}{{60}}hr$

Total time for complete journey is given by $T$ that is
$T = {t_1} + {t_2}$

Put values from above
$ \Rightarrow T = \dfrac{x}{{40}} + \dfrac{x}{{60}}$
$ \Rightarrow $ $\dfrac{{2x + 3x}}{{120}}$

Therefore total time is
$T = \dfrac{x}{{24}}hr$
Now average velocity of an object is its total displacement divided by the total time taken. In other words, it is the rate at which an object changes its position from one place to another. Average velocity is a vector quantity.
${\text{avg velocity = }}\dfrac{{{\text{total distance}}}}{{{\text{total time}}}}$

Inserting values of distance and time
$v = \dfrac{{2x}}{{x/24}}$
So avg velocity is
$v = 48km{h^{ - 1}}$

Note: Always remember about the units of quantities. Don’t confuse SI units and cgs units. Always use a single system of units in a question. Moreover, double check for the formula of average velocity because sometimes we use other formulas that look like the mean formula.