Question

A 1000 kg automobile is moving along a straight highway at $10\,m/s$ . Another car with a mass of 2000kg and speed $10\,m/s$ is 30m ahead of the first. The position of the center of mass of the system from the first is:
A) 0
B) 10 m
C) 20 m
D) 30 m

Answer 1

Hint : In this solution, we will make use of the fact that the cars are moving with the same velocity so the centre of mass will have a fixed position with respect to the position of both the cars. To determine the position of the centre of mass, we will use the formula for the centre of mass for a system of particles.

Formula used: In this solution we will be using the following formula,
${x_{COM}} = \dfrac{{\sum\limits_{i = 1}^n {{m_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{m_i}} }}$ where ${x_{COM}}$ is the $x$ coordinate of the centre of mass, ${x_i}$ and ${m_i}$ are the $x$ -coordinate and mass of the i-th object respectively

Complete step by step answer
We’ve been given that a 1000 kg automobile is moving along a straight highway at $10\,m/s$ and another car with a mass of 2000kg and speed $10\,m/s$ is 30m ahead of the first car. Since both the cars are moving at the same speed, the center of mass will remain at the same position between the two cars. Let us assume the cars as point masses and that the first car is located at coordinates $(0,0)$ and since the second coordinate is 30m ahead of the first, the coordinate of the second car will be $(30,0)$ .
To find the center of mass, we can use the x-coordinate of coordinates of the car as:
${x_{COM}} = \dfrac{{\sum\limits_{i = 1}^n {{m_i}{x_i}} }}{{\sum\limits_{i = 1}^n {{m_i}} }}$
For our system, ${m_1} = 1000\,{\text{and}}\,{m_2} = 2000$ so the centre of mass is,
${x_{COM}} = \dfrac{{1000(0) + 2000(30)}}{{1000 + 2000}}$
$\Rightarrow {x_{COM}} = 20\,m$
Hence the centre of mass lies at a distance of 20 m on the x-axis which is where the first car is located. Hence the position of the centre of mass is also 20 m which corresponds to option (B).

Note
Since the two cars are moving with the same velocity, the centre of mass will always lie at a distance of 20m from the first car. Since we want to find the distance of the centre of mass from the first car, it is beneficial to assume the coordinates of the first car as the origin as it will simplify our calculations.