91Ó°ÊÓ

Given that a 10.0 g marble slides to the left at a speed of 0.400 m/s on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0-g marble sliding to the right at a speed of 0.200 m/s.

Mass of smaller marble A${\mathrm{m}}_1=10\mathrm{g}$

Mass of larger marble B${\mathrm{m}}_2=30\mathrm{g}$

Initial speed of smaller marble A ${\mathrm{u}}_1=0.4\mathrm{m}/\mathrm{s}$${\mathrm{u}}_1=0.4\mathrm{m}/\mathrm{s}$

Initial speed of larger marble Brole="math" localid="1665043003365" ${\mathrm{u}}_2=0.2\mathrm{m}/\mathrm{s}$

After collision, let${\mathrm{v}}_1$and${\mathrm{v}}_2$be speeds of smaller and larger marble respectively.

During collision, momentum is conserved.

So, Initial momentum of marbles = final momentum of marbles$$\begin{gathered} {\mathrm{m}}_1{\mathrm{u}}_1+{\mathrm{m}}_2{\mathrm{u}}_2={\mathrm{m}}_1{\mathrm{v}}_1+{\mathrm{m}}_2{\mathrm{v}}_2 \\ \left(0.03\mathrm{kg}\right)\left(0.2\mathrm{m}/\mathrm{s}\right)+\left(0.01\mathrm{kg}\right)\left(-0.4\mathrm{m}/\mathrm{s}\right)=0.03\mathrm{kg}\left({\mathrm{v}}_1\right)+0.01\mathrm{kg}\left({\mathrm{v}}_2\right) \\ 0.2=3{\mathrm{v}}_1+{\mathrm{v}}_2 \end{gathered}$$

Since it is an elastic collision, therefore, from relative velocity equation

$$\begin{gathered} {\mathrm{v}}_2-{\mathrm{v}}_1=-\left({\mathrm{u}}_2-{\mathrm{u}}_1\right) \\ {\mathrm{v}}_2-{\mathrm{v}}_1=\left(-0.4\mathrm{m}/\mathrm{s}-0.2\mathrm{m}/\mathrm{s}\right) \\ 0.6={\mathrm{v}}_2-{\mathrm{v}}_1 \end{gathered}$$

Solving the two equations gives

$$\begin{gathered} {\mathrm{v}}_2=0.5\mathrm{m}/\mathrm{s} \\ {\mathrm{v}}_1=-0.1\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the 30-g marble is moving to the left at 0.1 m/s and 10-g marble is moving to the right at 0.5 m/s.

For smaller marble, change in momentum is

$$\begin{gathered} {\mathrm{m}}_1{\mathrm{v}}_1-{\mathrm{m}}_1{\mathrm{u}}_1=\left(0.03\mathrm{kg}\right)\left(-0.1\mathrm{m}/\mathrm{s}-0.2\mathrm{m}/\mathrm{s}\right) \\ =-0.009\mathrm{kgm}/\mathrm{s} \end{gathered}$$

Change in momentum for smaller marble is -0.009 Kg m/s.

For larger marble, change in momentum is

$$\begin{gathered} {\mathrm{m}}_2{\mathrm{v}}_2-{\mathrm{m}}_2{\mathrm{u}}_2=\left(0.01\mathrm{kg}\right)\left(-0.5\mathrm{m}/\mathrm{s}-\left(-0.4\mathrm{m}/\mathrm{s}\right)\right) \\ =-0.009\mathrm{kgm}/\mathrm{s} \end{gathered}$$

Change in momentum for larger marble is 0.009 Kg m/s.

The change in momentum have same magnitude but opposite sign.

For smaller marble, change in kinetic energy is

$$\begin{gathered} \frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2-\frac{1}{2}{\mathrm{m}}_1{\mathrm{u}}_1^2=\frac{1}{2}\left(0.03\mathrm{kg}\right)\left({\left(0.1\mathrm{m}/\mathrm{s}\right)}^2-{\left(0.2\mathrm{m}/\mathrm{s}\right)}^2\right) \\ =-4.5×{10}^{-4}\mathrm{J} \end{gathered}$$

Change in kinetic energy for smaller marble is $-4.5×{10}^{-4}\mathrm{J}$.

For larger marble, change in kinetic energy is

$$\begin{gathered} \frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2-\frac{1}{2}{\mathrm{m}}_2{\mathrm{u}}_2^2=\frac{1}{2}\left(0.01\mathrm{kg}\right)\left({\left(0.5\mathrm{m}/\mathrm{s}\right)}^2-{\left(0.4\mathrm{m}/\mathrm{s}\right)}^2\right) \\ =4.5×{10}^{-4}\mathrm{J} \end{gathered}$$

Change in kinetic energy for larger marble is$4.5×{10}^{-4}$ J

The change in kinetic energy have same magnitude but opposite sign.