Question

Two objects of masses $200g$ and $500g$ having velocities of $10m/s \widehat{i}$ and $(3\widehat{i}+5\widehat{j})m/s$ respectively. The velocity of their center of mass is:
A) $5\widehat{i} - 25\widehat{j}$
B) $\dfrac{5}{7}\widehat{i}-25\widehat{j}$
C) $5\widehat{i}+\dfrac{25}{7}\widehat{j}$
D) $25\widehat{i}-\dfrac{5}{7}\widehat{j}$

Answer 1

We must know the concepts of center of mass and conservation of momentum, as these two are necessary for solving this question. We can also just directly look at the options and mark the correct answer (see note for this).

Complete step by step solution:
Definition of center of mass: If a system of mass is confined to a single mass with corresponding mass distribution, the point at which the gravitational center of the mass is located, is called the center of mass.

Conservation of momentum: The initial momentum and the final momentum of a system remains the same. In other words, the total momentum of a system remains conserved.
Given the two masses with respective masses, we can just find the momentum of each and conserve the total momentum.
Let A and B be the two blocks respectively.
We know that, $1kg=1000g$,
${{M}_{A}}=200g=0.2kg$ and ${{M}_{B}}=500g=0.5kg$
${{V}_{A}}=10m/s \widehat{i}$ and ${{V}_{B}}=(3\widehat{i}+5\widehat{j})m/s$
Initial Momentum: ${{M}_{A}}{{V}_{A}}+{{M}_{B}}{{V}_{B}}=0.2\times 10\widehat{i}+0.5\times \left( 3\widehat{i}+5\widehat{j} \right)$
$\Rightarrow {{M}_{i}}=3.5\widehat{i}+2.5\widehat{j}$
Final Momentum: ${{M}_{total}}{{V}_{cm}}=0.7{{V}_{cm}}$
Applying conservation of momentum:
${{M}_{i}}={{M}_{total}}{{V}_{cm}}$
$\Rightarrow 3.5\widehat{i}+2.5\widehat{j}=0.7{{V}_{cm}}$
$\Rightarrow {{V}_{cm}}=5\widehat{i}+\dfrac{25}{7}\widehat{j}$
Hence, the velocity of the center of mass is:
${{V}_{cm}}=5\widehat{i}+\dfrac{25}{7}\widehat{j}$

Therefore, option (C) is correct.

Note: For this question, the two masses were having all positive velocities only. So, we can look at the option that contains only all positive velocities as the center of mass will also be traveling with the positive velocity only. But this might not always work as there might be more than one option that might contain all positive velocity. Check the units and convert them into the Standard unit or any other preferred unit according to the question. We must always convert the unit into a single unit system so that we don’t make any calculation errors.