Question

1.3. Two smooth spheres, each of mass m and radius r, are motionless and touching when a third sphere of mass m, radius r, and velocity vo strikes them in perfectly elastic impact, as shown. Solve for the velocities of the three spheres after impact. Introduction to particle dynamics 0-084 Figure P 1.3.

Answer 1

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In the above diagram, the line of action of internal forces, pass through the center of each sphere, as shown. Therefore the final velocities of ball 2&3 is also in the same direction. However, due to symmetry, the y-component of the forces on ball 1, cancel out and produce a net x-direction force. Therefore final velocity of ball 1, shall remain in the x-direction.

Applying conservation of momentum, in x-direction:

$\small m_1u_1=m_1v_1+m_2v_2\cos{45\degree}+m_3v_3\cos{45\degree}$

since all masses are equal,

-------------------------------------------------(1)   

+ 1 = IT + گان

Applying conservation of momentum, in y-direction:

------------------------(2)

mque sin 45º = m3u3 sin 45° → = v3

Applying conservation of energy:

$\small \frac{1}{2}m_1{u_1}^2=\frac{1}{2}m_1{v_1}^2+\frac{1}{2}m_2{v_2}^2+\frac{1}{2}m_3{v_3}^2$

$\small {u_1}^2={v_1}^2+{v_2}^2+{v_3}^2$----------------------------------------------------(3)

From equations 1&2;

$\small u_1=v_1+v_2\sqrt2$

$\small \rightarrow u_1-v_2\sqrt2=v_1$---------------------------------------------------------(4)

From equations 2, 3 &4;

$\small {u_1}^2={(u_1-v_2\sqrt2)}^2+2{v_2}^2$

$\small -2\sqrt2 v_2u_1+4{v_2}^2=0$

$\small v_2=\frac{u_1}{\sqrt2}$

Therefore,

$\small v_3=v_2=\frac{u_1}{\sqrt2}$

Therefore,

$\small v_1=u_1-\frac{u_1}{\sqrt2}\sqrt2=0$