![](\"resources/question-resources/collision.png\")
We can draw a diagram.
\n
The impulse applied to both balls is $I = \\var{impulse} \\, \\mathrm{Ns}$, and the speeds after the collision are $v_1 = \\var{A_speed} \\, \\mathrm{ms^{-1}}$ and $v_2 = \\var{B_speed} \\, \\mathrm{ms^{-1}}$ for the balls $A$ and $B$ respectively.
\nTo calculate $u_1$, the speed of ball $A$ before the collision, we use the equation $I = m_1v_1 - m_1u_1$, where $m_1$ is the mass of ball $A$, which is $\\var{A_mass} \\, \\mathrm{kg}$. We resolve in the direction of the impulse, therefore the signs of $v_1$ and $u_1$ will be reversed.
\n\\begin{align}
I & = m_1v_1 - m_1u_1, \\\\
\\var{impulse} & = \\var{A_mass} (v_1 - u_1), \\\\
\\frac{\\var{impulse}}{\\var{A_mass}} & = - \\var{A_speed} - (- u_1), \\\\
u_1 & = \\frac{\\var{impulse}}{\\var{A_mass}} + \\var{A_speed}, \\\\
& = \\var{siground( impulse/A_mass + A_speed,3)} \\, \\mathrm{ms^{-1}}.
\\end{align}
The magnitude of the speed of $A$ before the collision is $\\var{siground( impulse/A_mass + A_speed,3)} \\, \\mathrm{ms^{-1}}$.
\nTo calculate $u_2$, the speed of ball $B$ before the collision we use the equation $I = m_2v_2 - m_2u_2$, where $m_2$ is the mass of ball $B$, which is $\\var{B_mass} \\, \\mathrm{kg}$. We resolve in the direction of the impulse shown in the diagram, which is the same as the direction of the speeds.
\n\\begin{align}
I & = m_2v_2 - m_2u_2, \\\\
\\var{impulse} & = \\var{B_mass} ( \\var{B_speed} - u_2), \\\\
\\frac{\\var{impulse} }{ \\var{B_mass}} & = \\var{B_speed} - u_2, \\\\
u_2 & = \\var{B_speed} - \\frac{\\var{impulse} }{ \\var{B_mass}}, \\\\
& = \\var{siground( B_speed - (impulse/B_mass),3)} \\, \\mathrm{ms^{-1}}.
\\end{align}
If this is positive it means the direction of ball $B$ we assumed is correct; if it is negative it means the ball was originally travelling in the other direction. However we were asked to find the magnitude of the speed so we take our answer as $\\var{B_before_speed} \\, \\mathrm{ms^{-1}}$.
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\nAfter the collision, the $A$ travels at $\\var{A_speed} \\mathrm{ms^{-1}}$ and $B$ travels at $\\var{B_speed} \\mathrm{ms^{-1}}$. The magnitude of the impulse of $A$ on $B$ is $\\var{impulse} \\mathrm{Ns}$.
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