Question

Two bodies P and Q of mass 5 kg and 10 kg respectively are initially at rest. P and Q move towards each other under the mutual force of attraction. At the instant when the speed of P is 10 m/sec and Q is 20 m/sec then determine the speed of the center of mass of the system? 

1. 5 m/sec in direction of P
2. 10 m/sec in direction of P
3. 5 m/sec in direction of Q
4. 10 m/sec in direction of Q

Answer 1

Correct Answer - Option 4 : 10 m/sec in direction of Q

CONCEPT:

• The Center of mass is the location of particles inside the system where the total mass of the system can be considered to be concentrated.
• When the system of the particle is moving, the center of mass moves along with it.
• The velocity of the center of mass is given by the sum of the momentum of each particle divided by the total mass of the system
• The mathematical equation for a two-particle system is given by 

\(⇒ V_{cm}=\frac{m_{1}v_{1}+m_{2}v}{m_{1}+m_{2}}\)

Where m₁ and m₂ are mass of bodies in kg, and v₁ and v₂ represent velocity in m/sec.

CALCULATION:

Given - m₁ = 5kg, m₂ = 10kg, v₁ = 10m/sec, and v₂ = -20 m/sec (because both are moving in opposite direction)

• The speed of the center of mass of the system

\(⇒ V_{cm}=\frac{m_{1}v_{1}+m_{2}v}{m_{1}+m_{2}}\)

\(⇒ V_{cm}=\frac{10\times 5+10\times (-20))}{10+5}\)

\(⇒ V_{cm}=\frac{-150}{15}=-10 m/s\)

⇒ v_CM = -10m/sec

• The velocity of the center of mass is 10m/sec in direction of Q. 
• Hence option (4) is correct.