Complete Question
The complete question is shown on the first uploaded image
Answer:
The expression for the velocity of the sixth ball is
[tex]v_2 = \frac{u_1 sin\theta }{sin \phi cos\theta + cos \phi sin\theta}[/tex]
Explanation:
From the question we are told that
The mass of the 8 ball is [tex]m_1 = 0.5 \ kg[/tex]
The initial velocity of the 8 ball is [tex]u_1 = 3.7 \ m/s[/tex]
The initial velocity of the 6 ball is [tex]u_2 = 0 \ m/s[/tex]
The mass of the 6 ball is [tex]m_2 = 0.5 \ kg[/tex]
Generally from the law of momentum conservation is
[tex]m_1 *u_1 + m_2 * u_2 = m_1 * v_1 +m_2 *v_2[/tex]
Now along the y axis
[tex]u_1 =u_y_1 = 0 \ m/ s[/tex]
[tex]u_2 =u_y_2= 0 \ m/ s[/tex]
[tex]v_1 =v_y_1= v_1 sin \theta \ m/ s[/tex]
[tex]v_2 =v_y_2= v_2 sin \phi \ m/ s[/tex]
So
[tex]m_1 *0 + m_2 * 0 = m_1 * (v_1 sin \theta) +m_2 *(v_2 sin \phi )[/tex]
=> [tex]v_1 = \frac{v_2 sin \phi }{ sin \theta}[/tex]
Now along the x axis
[tex]u_1 =u_x_1 = u_1 \ m/ s[/tex]
[tex]u_2 =u_2_2= 0 \ m/ s[/tex]
[tex]v_1 =v_x_1= v_1 cos \theta \ m/ s[/tex]
[tex]v_2 =v_x_2= v_2 cos \phi \ m/ s[/tex]
So
[tex]m_1 *u_1 + m_2 * 0 = m_1 *(v_1 cos \theta) +m_2 *(v_2 cos \phi)[/tex]
=>[tex]m_1 *u_1 = m_1 *(v_1 cos \theta) +m_2 *(v_2 cos \phi)[/tex]
substituting for [tex]v_1[/tex] in the above equation
=>[tex]m_1 *u_1 = m_1 *([ \frac{v_2 sin \phi }{ sin \theta}] cos \theta) +m_2 *(v_2 cos \phi)[/tex]
=>[tex]m_1 *u_1 = m_1 *([ \frac{v_2 sin \phi }{ sin \theta}] cos \theta) +m_2 *(v_2 cos \phi)[/tex]
given that [tex]m_1 = m_2 = m[/tex]
=>[tex]m *u_1 = m *([ \frac{v_2 sin \phi }{ sin \theta}] cos \theta) +m *(v_2 cos \phi)[/tex]
=>[tex]m *u_1 = m [([ \frac{v_2 sin \phi }{ sin \theta}] cos \theta) + (v_2 cos \phi)][/tex]
=>[tex]u_1 = ([ \frac{v_2 sin \phi }{ sin \theta}] cos \theta) + (v_2 cos \phi)[/tex]
=>[tex]u_1 = v_2 sin\phi cot \theta + v_2 cos\phi [/tex]
[tex]v_2 = \frac{u_1 }{sin\phi cot\theta + cos\phi }[/tex]
converting [tex]cot \phi[/tex] back to [tex]\frac{cos \theta}{sin\theta }[/tex]
So
[tex]v_2 = \frac{u_1 }{sin\phi [\frac{cos \theta}{sin\theta }] + cos\phi }[/tex]
multiply through by [tex]\frac{sin \theta}{sin \theta }[/tex]
[tex] \frac{sin \theta}{sin \theta } *v_2 = \frac{sin \theta}{sin \theta } * \frac{u_1 }{sin\phi [\frac{cos \theta}{sin\theta }] + cos\phi }[/tex]
=> [tex]v_2 = \frac{u_1 sin\theta }{sin \phi cos\theta + cos \phi sin\theta}[/tex]
