First slide

Motion of centre of mass

Question

Two particles of equal mass have velocities $\vec{v_{1}} = 4 \hat{i}$ and $\vec{v_{2}} = 4 \hat{j}$ m/sec. First particle has an acceleration $\vec{a_{1}} = (5 \hat{i} + 5 \hat{j}) m s^{- 2}$ while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

Moderate

A

Straight line
B

Parabola
C

Circle
D

Ellipse

Solution

At any time 't',
co-ordinates of CM = ( $x_{c m}, y_{c m}$ ) $x_{c m} = \frac{m (4 t + \frac{1}{2} . 5 . t^{2})}{m + m} = 2 t + \frac{5 t^{2}}{4}$
similarly $y_{c m} = \frac{m (0 + \frac{1}{2} . 5 . t^{2}) + m (4 t)}{m + m} = \frac{5 t^{2}}{4} + 2 t$
Here $x_{c m} = y_{c m}$ for all 't'
Hence, the path of CM is straight line passing through origin making angle $45^{0}$

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