The Linear Momentum of a System of Particles — Question 3

Question 3

The frame in which total momentum vanishes

A system of mass $8.0\,\mathrm{kg}$ has total laboratory momentum $(24,-16)\,\mathrm{kg\,m/s}$.

1. Find the velocity of an inertial frame in which its total momentum is zero at that instant.

2. If the system later experiences a constant force $(0,8)\,\mathrm N$ for $2\,\mathrm s$, is its momentum still zero in that same frame?

Question 3 - Solution

The frame in which total momentum vanishes

(a) Derivation and calculation

Under a fixed inertial-frame shift, $$\vec P^{\prime}=\sum_i m_i(\vec v_i-\vec U)=\vec P-M\vec U.$$ $$\vec P_i^{\prime}=\vec0\Rightarrow\boxed{\vec U=\frac{(24,-16)}8=(3,-2)\ \mathrm{m/s}}.$$

(b) Result and interpretation

The frame velocity remains fixed after the force begins: $$\vec P_f^{\prime}=\vec P_i^{\prime}+\int_0^2(0,8)\,dt =\boxed{(0,16)\ \mathrm{kg\,m/s}}.$$ A frame that continuously follows the accelerating center would no longer be inertial.

Physical check

The same external impulse changes total momentum by the same amount in both inertial frames.