Answer:
a) mass of unknown nucleus = 0.04245 mp, where mp is the proton mass
b) Speed of the unknown nucleus = (7.067 x 10^7) m/s
Explanation:
Considering the initial conditions, the observed collisions are ellastic, i.e, the total kinetic energy are conserved. The proton's mass will refer as [tex]m_{p}[/tex].
(a)
Total kinetic energy conservation
[tex]\frac{1}{2}m_{p}v_{p_0}^{2}+\frac{1}{2}m_{u}v_{u_o}^{2}=\frac{1}{2}m_{p}v_{p_f}^{2}+\frac{1}{2}m_{u}v_{u_f}^{2}[/tex]
where [tex]v_{u_o}[/tex] represents the initial velocity of the unknown element, [tex]m_{u}[/tex] the mass of the unknown element, and [tex]v_{u_f}[/tex] the final velocity of the unknown element
Linear momentum conservation
[tex]m_{p}v_{p_0}+m_{u}v_{u_o}=m_{p}v_{p_f}+m_{u}v_{u_f}[/tex]
Using the initial speed of the target nucleus (unknown) is negligible, i.e, its speed is zero. Thereby, using the relation of linear momentum conservation given above, it is possible to find an expression of the final speed of the unknown nucleus in terms of its mass, which can be inserted in the relation of the kinetic energy conservation to obtain the value of the mass of the unknown elements, as follows;
[tex]m_{u}v_{u_f}=m_{p}v_{p_0}-m_{p}v_{p_f}\\\\v_{u_f}=\frac{m_{p}(v_{p_0}-v_{p_f})}{m_{u}}[/tex]
Substituting this expression in the relation of total kinetic energy conservation,
[tex]m_{p}(v^{2}_{p_0}-v^{2}_{p_f})={m_{u}}v^{2}_{u}_{f}[/tex]
Then,
[tex]m_{p}(v^{2}_{p_0}-v^{2}_{p_f})={m_{u}}\frac{m^{2}_{p}(v_{p_0}-v_{p_f})^{2}}{m^{2}_{u}}\\\\m_{u}= \frac{m_{p}(v_{p_0}-v_{p_f})^{2}}{(v^{2}_{p_0}-v^{2}_{p_f})}[/tex]
Replacing the given data
[tex]m_{u}= \frac{m_{p}(3.6x10^{7}-3.3x10^{7})^{2}}{((3.6x10^{7})^{2}-(3.3x10^{7})^{2})}[/tex]
Then,
[tex]m_{u}=0.04245m_{p}[/tex]
(b) Using the relation of the final speed from linear momentum conservation and the above result, the speed of the unknown nucleus is calculated
[tex]v_{u_f}=\frac{m_{p}(v_{p_0}-v_{p_f})}{m_{u}}\\\\v_{u_f}=\frac{m_{p}(3.6x10^{7}-3.3x10^{7})}{0.04245m_{p}}\\\\v_{u_f}=7.067x10^{7} m/s[/tex]
