Question

Consider a system of two identical particles. One of the particles is at rest and the other has an acceleration a. The centre of mass has an acceleration
(a) zero
(b) $\frac{1}{2}\overrightarrow{a}$
(c) $\overrightarrow{a}$
(d) $2\overrightarrow{a}$

Answer 1

(b) $\frac{1}{2}\overrightarrow{a}$

Acceleration of centre of mass of a two-particle system is given as,
${\overrightarrow{a}}_{cm}=\frac{m_1{{\displaystyle \overrightarrow{a}}}_1+m_2{\displaystyle \overrightarrow{a_2}}}{m_1+m_2}$ ...(1)

According to the question,

$$\begin{gathered} m_1=m_2=m \\ {a}_1=0 \\ {a}_2=a \end{gathered}$$

Substituting these values in equation (1), we get:
${\overrightarrow{a}}_{cm}=\frac{m\times 0+m{\displaystyle \overrightarrow{a}}}{2m}=\frac{1}{2}\overrightarrow{a}$