Question

A $50\mathrm{g}$marble moving at $2.0\mathrm{m}/\mathrm{s}$ strikes a $20\mathrm{g}$ marble at rest. What is the speed of each marble immediately after the collision?

Short answer

Marvel of mass $50\mathrm{g}$has speed $0.86\mathrm{m}/\mathrm{s}$.

Marvel of mass$20\mathrm{g}$has speed$2.9\mathrm{m}/\mathrm{s}.$

Step-by-step solution

Given information

We have given,

Masses of marvel =$50\mathrm{g}\mathrm{and}20\mathrm{g}.$

Speed of marvels =$2\mathrm{m}/\mathrm{s}\mathrm{and}0\mathrm{m}/\mathrm{s}.$

We have to find their final velocity after collision.

Simplify

The given Collison is a elastic collision.

In the elastic Collison the momentum and energy of the system is conserved.

Then conservation of momentum is,

${\mathrm{P}}_{\mathrm{i}}={\mathrm{P}}_{\mathrm{f}}$

Let us consider marvels as marvel 1 of $50\mathrm{g}$and marvel 2 of $20\mathrm{g}.$

Then,

$$\begin{gathered} {\mathrm{P}}_{\mathrm{i}}={\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2 \\ {\mathrm{P}}_{\mathrm{i}}=0.05\mathrm{kg}\times 2\mathrm{m}/\mathrm{s}+0 \\ {\mathrm{P}}_{\mathrm{i}}=0.1\mathrm{kg}\mathrm{m}/\mathrm{s} \end{gathered}$$

and

${\mathrm{P}}_{\mathrm{F}}={\mathrm{m}}_1{\mathrm{v}}_{1\mathrm{f}}+{\mathrm{m}}_2{\mathrm{v}}_{2\mathrm{f}}..........\left(1\right)$

Then using conservation of energy,

localid="1649676035847" $$\begin{gathered} \mathrm{K}.{\mathrm{E}}_{\mathrm{I}}=\mathrm{K}.\mathrm{E}.{\mathrm{F}}_{\mathrm{F}} \\ \frac{{\mathrm{m}}_1{\mathrm{v}}_1^2+{\mathrm{m}}_2{\mathrm{v}}_2^2}{2}=\frac{{\mathrm{m}}_1{\mathrm{v}}_{1\mathrm{f}}^2+{\mathrm{m}}_2{\mathrm{v}}_{2\mathrm{f}}^2}{2} \\ (0.05)\left(4\right)={\mathrm{m}}_1{\mathrm{v}}_{1\mathrm{f}}^2+{\mathrm{m}}_2{\mathrm{v}}_{2\mathrm{f}}^2........................\left(2\right) \end{gathered}$$

Using equation (1) AND (2),

$$\begin{gathered} {\mathrm{v}}_{1\mathrm{f}}=0.86\mathrm{m}/\mathrm{s} \\ {\mathrm{v}}_{2\mathrm{f}}=2.9\mathrm{m}/\mathrm{s} \end{gathered}$$