Q: An object A is dropped from the top of a 30 m high building and at the same moment another object B is projected vertically upwards with an initial speed of 15 m/s from the base of the building. Mass of the object A is 2kg while mass of the object B is 4 kg. Find the maximum height about the ground level attained by the centre of mass of A and B system (take g = 10 m/s2)

Sol: m1 = 4 kg, m2 = 2 kg. Initially 4 kg is on the ground, therefore x1 = 0 and 2 kg is on top of the building,

therefore x2 = 30m.

$\large x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

$\large = \frac{0 + 2 \times 30}{4 + 2} = 10 m $

Initial velocity of CM,

$\large u_{cm} = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}$

$\large u_{cm} = \frac{4 \times 15 + 0}{4+2} $

=10 m/s upward

acceleration of CM,

a_{CM} = g = 10m/s^{2} downwards.

Maximum height attained by CM from initial position,

$\large H_{cm} = \frac{u_{cm}^2}{2 g} = \frac{10^2}{20} = 5 m$