(a) The final velocity of the ball is 50.59 m/s at an angle [tex]2.6 ^\circ[/tex] with the horizontal.
(b) The collision is inelastic.
(c) The spin of the ball is converted to linear kinetic energy in the collision due to the slipping of the ball.
Collision
From the figure (given in attachment), we can write the equation applying the law of conservation of momentum.
First consider the vertical motion of pin and the ball.
- There is no initial velocities in the vertical direction for both the pin and the ball.
- [tex]0=(m_Bv_B\,sin\, \alpha)_{final} - (m_P\,v_P\,sin\, 85^\circ)_{final}[/tex]
Substituting the known values, we get
- [tex](v_B sin \alpha)_{final}=\frac{0.850\,kg \times 15.0\,m/s \times 0.99}{5.50\,kg} =2.295\,m/s[/tex]
Now consider the horizontal motion.
- The ball has a horizontal initial velocity while the pin is at rest.
- [tex]m_Bv_B = (m_Bv_B\,cos\, \alpha)_{final} - (m_P\,v_P\,cos\, 85^\circ)_{final}[/tex]
Substituting the given values we get;
- [tex](v_B\,cos\, \alpha)_{final} =\frac{(5.50\times 9.0 \,m/s)+(0.85\,kg\times\,15.0\,m/s\times cos\, 85^\circ)}{5.50\,kg} =50.61\,m/s[/tex]
Dividing the horizontal and vertial components of final velocities of pinn and the ball, we get;
- [tex]\frac{(v_B \,sin\, \alpha)_{final}}{(v_B \,cos\, \alpha)_{final}} =\frac{2.295}{50.61}[/tex]
- [tex]tan \, alpha = 0.0453[/tex]
- [tex]\implies \alpha = tan^{-1}(0.0453)=2.59 ^\circ \approx 2.6 ^\circ[/tex]
We know that, from previous steps;
[tex](v_B \,sin\, \alpha)_{final} ={2.295}\,m/s[/tex]
- [tex]\implies (v_B)_{final} = \frac{2.295\, m/s}{sin\, \ 2.6 ^\circ} =50.59\,m/s[/tex]
(b) For the collision to be elastic, the kinetic energies before and after collision must be equal.
[tex]KE_{initial}= \frac{1}{2} m_B(v_B)^2[/tex]
- [tex]KE_{initial}= \frac{1}{2} (5.50\,kg)(9\,m/s)^2=222.75\,J[/tex]
[tex]KE_{final}= \frac{1}{2} m_B(v_B)_{final}^2+\frac{1}{2} m_B(v_P)_{final}^2[/tex]
- [tex]KE_{final}= \frac{1}{2} (5.50\,kg)(50.59\,m/s)^2+\frac{1}{2} (0.850\,kg)(15\,m/s)^2=7044.58\,m/s[/tex]
Therefore, the collision is inelastic.
(c) Just after the collision the ball rotates so that the angular velocity of the ball will be greater than its linear velocity. This causes slipping that accelerates the ball, which results in the conversion of rotational kinetic energy to linear kinetic energy.
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