Answer:
The final speed for the mass 2m is
and the final speed for the mass 9m is
.
The angle at which the particle 9m is scattered is
with respect to the - y axis.
Explanation:
In an elastic collision the total linear momentum and the total kinetic energy is conserved.
<u>Conservation of linear momentum:</u>
Because the linear momentum is a vector quantity we consider the conservation of the components of momentum in the x and y axis.
The subindex 1 will refer to the particle 9m and the subindex 2 will refer to the particle 2m
![\vec{p}=m\vec{v}](https://tex.z-dn.net/?f=%5Cvec%7Bp%7D%3Dm%5Cvec%7Bv%7D)
![p_{xi} =p_{xf}](https://tex.z-dn.net/?f=p_%7Bxi%7D%20%3Dp_%7Bxf%7D)
In the x axis before the collision we have
![p_{xi}=9m\ v_{i} - 2m\ v_{i}](https://tex.z-dn.net/?f=p_%7Bxi%7D%3D9m%5C%20v_%7Bi%7D%20-%202m%5C%20v_%7Bi%7D)
and after the collision we have that
![p_{xf} =9m\ v_{1x}](https://tex.z-dn.net/?f=p_%7Bxf%7D%20%3D9m%5C%20v_%7B1x%7D)
In the y axis before the collision ![p_{yi} =0](https://tex.z-dn.net/?f=p_%7Byi%7D%20%3D0)
after the collision we have that
![p_{yf} =9m\ v_{1y} - 2m\ v_{2y}](https://tex.z-dn.net/?f=p_%7Byf%7D%20%3D9m%5C%20v_%7B1y%7D%20-%202m%5C%20v_%7B2y%7D)
so
![p_{xi} =p_{xf} \\7m\ v_{i} =9m\ v_{1x}\Rightarrow v_{1x} =\frac{7}{9}\ v_{i}](https://tex.z-dn.net/?f=p_%7Bxi%7D%20%3Dp_%7Bxf%7D%20%5C%5C7m%5C%20v_%7Bi%7D%20%3D9m%5C%20v_%7B1x%7D%5CRightarrow%20v_%7B1x%7D%20%3D%5Cfrac%7B7%7D%7B9%7D%5C%20v_%7Bi%7D)
then
![p_{yi} =p_{yf} \\0=9m\ v_{1y} -2m\ v_{2y} \\v_{1y}=\frac{2}{9} \ v_{2y}](https://tex.z-dn.net/?f=p_%7Byi%7D%20%3Dp_%7Byf%7D%20%5C%5C0%3D9m%5C%20v_%7B1y%7D%20-2m%5C%20v_%7B2y%7D%20%5C%5Cv_%7B1y%7D%3D%5Cfrac%7B2%7D%7B9%7D%20%5C%20v_%7B2y%7D)
<u>Conservation of kinetic energy:</u>
![\frac{1}{2}\ 9m\ v_{i} ^{2} +\frac{1}{2}\ 2m\ v_{i} ^{2}=\frac{1}{2}\ 9m\ v_{1f} ^{2} +\frac{1}{2}\ 2m\ v_{2f} ^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%5C%209m%5C%20v_%7Bi%7D%20%5E%7B2%7D%20%2B%5Cfrac%7B1%7D%7B2%7D%5C%202m%5C%20v_%7Bi%7D%20%5E%7B2%7D%3D%5Cfrac%7B1%7D%7B2%7D%5C%209m%5C%20v_%7B1f%7D%20%5E%7B2%7D%20%2B%5Cfrac%7B1%7D%7B2%7D%5C%202m%5C%20v_%7B2f%7D%20%5E%7B2%7D)
so
![\frac{11}{2}\ m\ v_{i} ^{2} =\frac{1}{2} \ 9m\ [(\frac{7}{9}) ^{2}\ v_{i} ^{2}+ (\frac{2}{9}) ^{2}\ v_{2y} ^{2}]+ m\ v_{2y} ^{2}](https://tex.z-dn.net/?f=%5Cfrac%7B11%7D%7B2%7D%5C%20m%5C%20v_%7Bi%7D%20%5E%7B2%7D%20%3D%5Cfrac%7B1%7D%7B2%7D%20%5C%209m%5C%20%5B%28%5Cfrac%7B7%7D%7B9%7D%29%20%5E%7B2%7D%5C%20v_%7Bi%7D%20%5E%7B2%7D%2B%20%28%5Cfrac%7B2%7D%7B9%7D%29%20%5E%7B2%7D%5C%20v_%7B2y%7D%20%5E%7B2%7D%5D%2B%20m%5C%20v_%7B2y%7D%20%5E%7B2%7D)
Putting in one side of the equation each speed we get
![\frac{25}{9}\ m\ v_{i} ^{2} =\frac{11}{9}\ m\ v_{2y} ^{2}\\v_{2y} =-1,51\ v_{i}](https://tex.z-dn.net/?f=%5Cfrac%7B25%7D%7B9%7D%5C%20m%5C%20v_%7Bi%7D%20%5E%7B2%7D%20%3D%5Cfrac%7B11%7D%7B9%7D%5C%20m%5C%20v_%7B2y%7D%20%5E%7B2%7D%5C%5Cv_%7B2y%7D%20%3D-1%2C51%5C%20v_%7Bi%7D)
We know that the particle 2m travels in the -y axis because it was stated in the question.
Now we can get the y component of the speed of the 9m particle:
![v_{1y} =\frac{2}{9}\ v_{2y} \\v_{1y} =-0,335\ v_{i}](https://tex.z-dn.net/?f=v_%7B1y%7D%20%3D%5Cfrac%7B2%7D%7B9%7D%5C%20v_%7B2y%7D%20%5C%5Cv_%7B1y%7D%20%3D-0%2C335%5C%20v_%7Bi%7D)
the magnitude of the final speed of the particle 9m is
![v_{1f} =\sqrt{v_{1x} ^{2}+v_{1y} ^{2} }](https://tex.z-dn.net/?f=v_%7B1f%7D%20%3D%5Csqrt%7Bv_%7B1x%7D%20%5E%7B2%7D%2Bv_%7B1y%7D%20%5E%7B2%7D%20%7D)
![v_{1f} =\sqrt{(\frac{7}{9}) ^{2}\ v_{i} ^{2}+(-0,335)^{2}\ v_{i} ^{2} }\Rightarrow \ v_{1f} =0,85\ v_{i}](https://tex.z-dn.net/?f=v_%7B1f%7D%20%3D%5Csqrt%7B%28%5Cfrac%7B7%7D%7B9%7D%29%20%5E%7B2%7D%5C%20v_%7Bi%7D%20%5E%7B2%7D%2B%28-0%2C335%29%5E%7B2%7D%5C%20v_%7Bi%7D%20%5E%7B2%7D%20%7D%5CRightarrow%20%5C%20v_%7B1f%7D%20%3D0%2C85%5C%20v_%7Bi%7D)
The tangent that the speed of the particle 9m makes with the -y axis is
![tan(\theta)=\frac{v_{1x} }{v_{1y}} =-2,321 \Rightarrow\theta=-66,68^{o}](https://tex.z-dn.net/?f=tan%28%5Ctheta%29%3D%5Cfrac%7Bv_%7B1x%7D%20%7D%7Bv_%7B1y%7D%7D%20%3D-2%2C321%20%5CRightarrow%5Ctheta%3D-66%2C68%5E%7Bo%7D)
As a vector the speed of the particle 9m is:
![\vec{v_{1f} }=\frac{7}{9} v_{i} \hat{x}-0,335\ v_{i}\ \hat{y}](https://tex.z-dn.net/?f=%5Cvec%7Bv_%7B1f%7D%20%7D%3D%5Cfrac%7B7%7D%7B9%7D%20v_%7Bi%7D%20%5Chat%7Bx%7D-0%2C335%5C%20v_%7Bi%7D%5C%20%5Chat%7By%7D)
As a vector the speed of the particle 2m is:
![\vec{v_{2f} }=-1,51\ v_{i}\ \hat{y}](https://tex.z-dn.net/?f=%5Cvec%7Bv_%7B2f%7D%20%7D%3D-1%2C51%5C%20v_%7Bi%7D%5C%20%5Chat%7By%7D)