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A particle moving in a straight line covers half the distance with a speed of $3m{s^{ - 1}}$. The other half of distance is covered with double speed $6m{s^{ - 1}}$. The average speed of the particle is:
A) $4 \cdot 5m{s^{ - 1}}$
B) $4m{s^{ - 1}}$
C) $9m{s^{ - 1}}$
D) $5m{s^{ - 1}}$

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: To calculate the average speed, we have to calculate the total distance covered and the total time taken to cover the distance. With that, we can calculate the average speed with the formula,
$s = \dfrac{D}{T}$
where D = total distance in metres and T = total time in seconds.

Complete step by step answer:
The average is defined as a number that expresses the central value, which is calculated by the sum of the values divided by the total number of values.
If a, b, c are three numbers, the average of these numbers is given by –
$A = \dfrac{{a + b + c}}{3}$
Speed is defined as the rate of change of distance per unit time.
$s = \dfrac{d}{t}$
If the speed is not constant and it keeps varying with time, we have to consider the average speed.
The average speed is used to understand the rate at which the object is moving. Mathematically, if the distance ${d_1}$ is covered in time ${t_1}$, distance ${d_2}$ is covered in time ${t_2}$ and distance ${d_3}$ is covered in time ${t_3}$, the average speed is given by –
$s = \dfrac{{{d_1} + {d_2} + {d_3}}}{{{t_1} + {t_2} + {t_3}}}$
Consider a particle moving in a straight line. Let $x$ be the total distance covered by the particle.
Half of the distance i.e. $\dfrac{x}{2}$ is covered with the speed of $3m{s^{ - 1}}$. The time taken to cover this distance is given by –
${t_1} = \dfrac{{{d_1}}}{{{s_1}}} = \dfrac{{\dfrac{x}{2}}}{3} = \dfrac{x}{6}$
The other half of the distance i.e. $\dfrac{x}{2}$ is covered with the speed of $6m{s^{ - 1}}$. The time taken to cover this distance is given by –
${t_2} = \dfrac{{{d_2}}}{{{s_2}}} = \dfrac{{\dfrac{x}{2}}}{6} = \dfrac{x}{{12}}$
Thus,
Total distance travelled = $x$
Total time taken, = ${t_1} + {t_2} = \dfrac{x}{6} + \dfrac{x}{{12}} = \dfrac{{3x}}{{12}} = \dfrac{x}{4}$
The average speed, $S = \dfrac{x}{{{t_1} + {t_2}}} = \dfrac{x}{{\dfrac{x}{4}}} = 4m{s^{ - 1}}$

Hence, the correct option is Option (B).

Note: While the average speed gives us an overall picture of how fast the object is moving, the actual speed of the object in its journey will be constantly changing every instant of time. This type of speed is called instantaneous speed and is given by:
$s = \mathop {\lim }\limits_{x \to 0} \dfrac{{\Delta x}}{{\Delta t}} = \dfrac{{dx}}{{dt}}$