Answer:
[tex]\frac{1}{2}\frac{(M_dV_0)^2}{(M_d+M_a)^2} = h[/tex]
Explanation:
First, we will use conservation of the linear momentum:
[tex]P_i =P_f[/tex]
so:
[tex]M_dv_0 = (M_d+M_a)V_s[/tex]
where [tex]M_d[/tex] is the mass of the dart, [tex]v_0[/tex] is the speed of the dart just before it strickes the apple, [tex]M_a[/tex] the mass of the apple and [tex]V_s[/tex] the velocity of the apple and the dart after the collition.
Then, solving for V_s:
[tex]V_s = \frac{M_dV_0}{M_d+M_a}[/tex]
now, using the conservation of energy:
[tex]E_i = E_f[/tex]
so:
[tex]\frac{1}{2}(M_d+M_a)V_s^2 = (M_a+M_d)gh[/tex]
where g is the gravity and h how high does the apple move upward.
Now, replacing [tex]V_s[/tex] and solving for h, we get:
[tex]\frac{1}{2}(M_d+M_a)(\frac{M_dV_0}{M_d+M_a})^2 = (M_a+M_d)h[/tex]
[tex]\frac{1}{2}\frac{(M_dV_0)^2}{(M_d+M_a)^2} = h[/tex]
