Problem $1$

$[$Note: This problem is algebraic intensive, but I will not penalize heavily for algebraic mistakes.$]$

Two particles of masses $m_1$ and $m_2$ collide on a frictionless surface, see Figure $1.$ Particle $m_1$ has an initial speed of $u_1$ in the $x-$direction and particle $m_2$ is initially stationary. After the collision particle $m_1$ has a speed $v_1$ with a trajectory that makes an angle ${}_1$ with the x$-$axis, and particle $m_2$ has a speed $v_2$ with a trajectory that makes an angle ${}_2$ with the $x-$axis.

a$)$ Determine the following kinematic qualities of the system in terms of $v_1,v_2,{}_1$ and ${}_2.$ Each entry is worth $3$ points, make sure to show your work by drawing the appropriate triangles.

a$.v_{1,x}=$

b$.v_{1,y}=$

c$.v_{2,x}=$

d$.v_{2,y}=$

b$)$ Write the equations for conservation of momentum in the $x$ and $y$ directions, each equation is worth $3$ points.

a$.$ x$-$direction:

b$.y-$direction:

c$)$ Using the $y-$direction equation you derived above, a$)$ Solve for the final speed of particle $1,v_1,$ in terms of $m_1,m_2,{}_1,{}_2$ and $v_2.($b$)$ Using the equation from $($a$)$ determine the relationship between $m_1,m_2,sin({}_1),$ and $sin({}_2)$ that must be true if the final speeds of both particles are equal $($i$.$e$.v_1=v_1).$

a$.v_1=$

b$.\frac{m_1}{m_2}=$

d$)$ Using the equation for $v_1$ derived in part $($c$),$ determine the equation for the final speed of particle $2,v_2,$ in terms of $m_1,m_2,{}_1,{}_2$ and $u_1.$

a$.v_2=$

e$)$ Use the equation for $v_2$ derived in part d$)$ to determine the equation for the final speed of particle $1,v_1,$ in terms of $m_1,m_2,{}_1,{}_2$ and $u_1.$

a$.v_1=$

f$)$ Using the following values:

i$.u_1=10\frac{(m)}{(s)}$

ii$.{}_1=45$

iii. ${}_2=30$

iv$.m_2=5kg$

Determine the value of $m_1$ that will make the collision elastic: $m_1=$

BEFORE COLLISION

vec$(u{)}_1{?}_{m_1.}{m}_2$

AFTER COLLISION

Problem $1$

$[$Note: This problem is algebraic intensive, but I will not penalize heavily for algebraic mistakes.$]$

Two particles of masses $m_1$ and $m_2$ collide on a frictionless surface, see Figure $1.$ Particle $m_1$ has an initial speed of $u_1$ in the $x-$direction and particle $m_2$ is initially stationary. After the collision particle $m_1$ has a speed $v_1$ with a trajectory that makes an angle ${}_1$ with the $x-$axis, and particle $m_2$ has a speed $v_2$ with a trajectory that makes an angle ${}_2$ with the $x-$axis.

a$)$ Determine the following kinematic qualities of the system in terms of $v_1,v_2,{}_1$ and ${}_2.$ Each entry is worth $3$ points, make sure to show your work by drawing the appropriate triangles.

a$.,v_{1,x}=$

b$.,v_{1,y}=$

c$.,v_{2,x}=$

d$.v_{2,y}=$

b$)$ Write the equations for conservation of momentum in the $x$ and $y$ directions, eac