Exercise 7,12 Q1 (2013) (k)Sketch, on the same diagram, the paths of three alpha particles, of the same energy, which are directed towards a fixed nucleus so they are deflected through (i) 10 rr , (ii) 90 or , and (iii) I80r, If the nucleus in (iii) is not fixed, what is the relative velocity of the two particles at the distance of closest possible approach?[5]

To sketch the paths of three alpha particles directed towards a fixed nucleus and deflected at angles of 10°, 90°, and 180°, we can follow these steps: 1. **Understanding the Setup**: - The alpha particles are positively charged and will experience a repulsive force when approaching the positively charged nucleus. - The deflection angles indicate how much the paths of the alpha particles will curve due to this repulsion. 2. **Sketching the Paths**: - **Path for 10° deflection**: This path will be slightly curved, indicating a small deflection from the original straight-line path towards the nucleus. The particle will come close to the nucleus but will not be significantly affected. - **Path for 90° deflection**: This path will show a more pronounced curvature, indicating that the alpha particle has been significantly deflected by the nucleus. It will approach the nucleus, experience a strong repulsive force, and then be deflected at a right angle. - **Path for 180° deflection**: This path will show that the alpha particle approaches the nucleus, gets repelled, and then reverses direction, effectively coming back along the line it approached from. 3. **Relative Velocity at Closest Approach**: - For the case where the nucleus is not fixed (as in the 180° deflection scenario), both the alpha particle and the nucleus will be in motion. - The relative velocity of the two particles at the distance of closest approach can be determined by considering conservation of momentum and energy. - If we denote the mass of the alpha particle as \( m \) and the mass of the nucleus as \( M \), and if the alpha particle has an initial velocity \( v \) directed towards the nucleus, the nucleus will also start moving due to the repulsion. - At the point of closest approach, the relative velocity \( v_{rel} \) can be expressed as: \[ v_{rel} = v + v_n \] where \( v_n \) is the velocity of the nucleus at that point. The exact value of \( v_n \) will depend on the initial conditions and the masses involved. 4. **Conclusion**: - The sketch will show three distinct paths for the alpha particles, with the angles of deflection clearly marked. - The relative velocity at the closest approach will depend on the initial conditions and the masses of the particles involved, but it will be the sum of the velocities of the alpha particle and the nucleus at that point. This explanation provides a conceptual framework for sketching the paths and understanding the dynamics involved in the scenario described.