Question

A point mass of $1\text{kg}$ collides elastically with a stationary point mass of $5\text{kg}$. After collision the $1\text{kg}$ mass reverses its direction and moves with a speed of $2{\text{ms}}^{-1}$.
Which of the following statement(s) is/are correct for the system of these two masses?

• A. Total momentum of the system after the collision is $3{\text{Kg ms}}^{-1}$
• B. Momentum of $5\text{kg}$ mass after the collision is $4{\text{Kg ms}}^{-1}$
• C. Total kinetic energy of the system is $4\text{J}$
• D. Kinetic energy of the center of mass is $0.75\text{J}$

Answer 1

Answer: (A), (D)

The correct options are
A Total momentum of the system after the collision is $3{\text{Kg ms}}^{-1}$
D Kinetic energy of the center of mass is $0.75\text{J}$




Given, $m_1=1\text{kg};m_2=5\text{kg}$

$u_1=u;u_2=0{\text{ms}}^{-1}$ and $v_1=2{\text{ms}}^{-1};v_2=v$

In collision, only internal forces are involved, hence momentum is conserved.

$P_i=P_f$

$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

$u=5v-2.......\left(1\right)$

Since collision is elstatic, $e=1$

$\therefore e=\frac{\text{velocities of separation}}{\text{velocities of approach}}=\frac{(v_2-v_1)}{(u_1-u_2)}$

$\Rightarrow 1=\frac{v+2}{u}$

$\Rightarrow u=v+2........\left(2\right)$

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

$u=3{\text{ms}}^{-1}$ and $v=1{\text{ms}}^{-1}$

Now, total moment of system is,

$P_T=m_1{v}_1+m_2{v}_2$

$=1\times (-2)+5\times 1=3{\text{kg ms}}^{-1}$

Now, total moment of $5\text{kg}$ after the collision is,

$P_{5kg}=m_2{v}_2=5{\text{kg ms}}^{-1}$

Total kinetic energy of the system,

$K{E}_T=\frac{1}{2}{m}_1{v}_1^2+\frac{1}{2}{m}_2{v}_2^2=2+2.5=4.5\text{J}$

Kinetic energy of the center of mass is,

$K{E}_{cm}=\frac{1}{2}\times m_{sys}\times v_{cm}^2$

$=\frac{1}{2}\times (1+5)\times {\left(\frac{(1\times u)+(5\times 0)}{1+5}\right)}^2$

$=\frac{1}{2}\times 6\times \frac{9}{36}=0.75\text{J}$

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

Why this question? Key concept: In an elastic collision total kinetic energy before and after collision is conserved.