Question

Two particles $\text{A}$ and $\text{B}$ of masses $1\text{kg}$ and $2\text{kg}$ respectively are projected with speed $u_A=200{\text{ms}}^{-1}$ and $u_B=50{\text{ms}}^{-1}$ in the directions as shown in figure. Initially they were $90\text{m}$ apart. They collide in the midair and stick with each other. Find the maximum height attained by the centre of mass of the system.
[Assume acceleration due to gravity to be constant as <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $g=10{\text{ms}}^{-2}$]

• A. $60\text{m}$
• B. $115.55\text{m}$
• C. $55.55\text{m}$
• D. $40\text{m}$

Answer 1

The correct option is B $115.55\text{m}$

Given:
$m_A=1\text{kg};m_B=2\text{kg}$
$u_A=200{\text{ms}}^{-1};u_B=50{\text{ms}}^{-1}$
and initial separation between them is,
$d=90\text{m}$

Using the relation for centre of mass of the system,

$r_{\text{com}}=\frac{m_A{r}_A+m_B{r}_B}{m_A+m_B}$


Taking $r_{\text{com}}$ as the origin we get,

$0=\frac{m_A(-r_A)+m_B{r}_B}{m_A+m_B}$

$m_A{r}_A=m_B{r}_B\Rightarrow 1\times r_A=2\times r_B$

$\Rightarrow r_A=2r_B.......\left(1\right)$

and from given data,

$r_A+r_B=90\text{m}.......\left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$, we get,

$r_A=60\text{m}$ and $r_B=30\text{m}$

i.e,. COM is at height $60\text{m}$ from the ground.

Further, velocity of the COM will be,

$u_{COM}=\frac{m_A{u}_A+m_B{u}_B}{m_A+m_B}$

$u_{COM}=\frac{\left(1\right)\left(200\right)-\left(2\right)\left(50\right)}{1+2}$

$\therefore u_{COM}=\frac{100}{3}{\text{ms}}^{-1}(\uparrow )$

Let $h$ be the height attained by COM beyond $60\text{m}$.

Using, kinematic equations of motion,

$v_{COM}^2=u_{COM}^2+2a_{COM}h$

$0={\left(\frac{100}{3}\right)}^2-\left(2\right)\left(10\right)h$

$h=\frac{(100{)}^2}{180}=55.55\text{m}$

Therefore, the maximum height attained by the centre of mass is,

$H=60+55.55=115.55\text{m}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.

Why this Question? This question is aimed at developing the student's approach towards solving the problems related to the motion of COM