Question

Can objects in a system have momentum while the momentum of the system is zero? Explain your answer.

Short answer

Yes, it true that objects in a system can have momentum while the momentum of the system is zero.

Step-by-step solution

Basics of Momentum

Momentum is equal to the force required to bring an item to a complete stop in a given amount of time when its mass is multiplied by its velocity. The total momentum of a group of objects is equal to the sum of their individual momenta.

\(P = mv\)

For any aligned of several objects

\(\begin{aligned}{P_{net}} &= {P_1} + {P_2} + {P_3} + {P_4}.............{P_n}\\{P_{net}} &= {m_1}{v_1} + {m_2}{v_2} + {m_3}{v_3} + .......{m_n}{v_n}\end{aligned}\)

Explaining that the objects in a system can have momentum while the momentum of the system is zero

To understand the above statement let’s consider an elastic collision of two bodies of similar mass m.There is a head-on collision between these two bodies and they are moving with the same speed,v but in opposite direction.

Linear momentum of bodies before collision

So, before collision, the linear momentum of body -1 is

\({P_1} = mv\)

the linear momentum of body-2 is

\({P_2} = m( - v) = - mv\)

The net initial momentum of the bodies

\(\begin{aligned}{P_{net}} &= {P_1} + {P_2}\\{P_{net}} &= mv + ( - mv)\\{P_{net}} &= mv - mv\\{P_{net}} &= 0\end{aligned}\)

Hence, the net initial momentum of the bodies is zero while individual body is having the momentum.

Linear momentum of bodies before collision

In elastic collision when two similar body collides to each other then after collision their velocities get exchanged so after collisionthe linear momentum of body -1 is

\({P_1} = - mv\)

the linear momentum of body-2 is

\({P_2} = mv\)

The net initial momentum of the bodies

\(\begin{aligned}{P_{net}} &= {P_1} + {P_2}\\{P_{net}} &= - mv + (mv)\\{P_{net}} &= - mv + mv\\{P_{net}} &= 0\end{aligned}\)

So, net final momentum of the system is zero while individual body has momentum.

From the above example it can easily undertand that in both the cases, before collision and after collision each body has its momentum but net momentum before collision and after collision is zero.

Hence, the objects in a system can have momentum while the momentum of the system is zero.