Question

A particle travels half of the total distance with a speed and next half with speed \[{{v}_{2}}\] along a straight line. Find the average speed of the particle?
1- \[\dfrac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]
2- \[\dfrac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}-{{v}_{2}}}\]
3- \[\dfrac{{{v}_{1}}+{{v}_{2}}}{2{{v}_{1}}{{v}_{2}}}\]
4- \[\dfrac{{{v}_{1}}-{{v}_{2}}}{2{{v}_{1}}{{v}_{2}}}\]

Answer 1

Hint: This is an example of motion in one dimension. Let us assume that total distance is 2d. So, one half of it will be d.
To find out the average speed we need to find out the total time taken by the body to cover the same.

Complete step by step answer:
For first half: distance covered is d
Speed= \[{{v}_{1}}\]
Distance= d
So, time taken = \[{{t}_{1}}=\dfrac{d}{{{v}_{1}}}\]
For second half: distance covered is d
Speed= \[{{v}_{2}}\]
Distance= d
So, time taken = \[{{t}_{2}}=\dfrac{d}{{{v}_{2}}}\]
So average speed= \[\dfrac{d+d}{{{t}_{1}}+{{t}_{2}}}\]
\[\Rightarrow \dfrac{d+d}{\dfrac{d}{{{v}_{1}}}+\dfrac{d}{{{v}_{2}}}}\]
\[\Rightarrow \dfrac{d+d}{\dfrac{d}{{{v}_{1}}}+\dfrac{d}{{{v}_{2}}}}=\dfrac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]

So, the correct answer is “Option 1”.

Additional Information:
Speed is defined as the ratio of distance to the time while velocity is defined as the ratio of displacement covered to the total time taken. Odometer is a measuring instrument which shares the space with the speedometer and is present in cars and bikes. The speedometer is the device used in a vehicle to measure the speed of that vehicle. Speedometer measures the instantaneous speed of the vehicle.

Note:
We should remember that distance is a scalar quantity and displacement is a vector quantity. Speed is the rate of change of distance and velocity is the rate of change of displacement. Average speed is equal to the total distance covered divided by the total time taken. All units are to be used to be taken in SI units.