When the bob is at athe horizontal position A, its height above B = length of the string =  ℓ . Therefore, potential energy at A = mgℓ . When the bob is released, it hits thewall at B and the enitre potentialenergy is converted into kinetic energy. If v is the velocity with which the bob hits wall, then

$\frac{1}{2}{\mathrm{mv}}^2=\mathrm{mg\ell }\text{\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}\mathrm{v}=\sqrt{2\mathrm{g\ell }}$                                   ... (i)

Speed of the bob after the first rebound,

v₁ = ev

Speed of the bob afte the second rebound,

v₂ = ev₁ = e²v               

Speed of the bob after n rebounds, v_n = eⁿv

If the ball rises to a position C at a height

h = BD afte n rebounds, then from the principle of conservation of energy, we have

$\frac{1}{2}{\mathrm{mv}}_{\mathrm{n}}^2=\mathrm{mgh}$

$\mathrm{h}=\frac{{\mathrm{v}}_{\mathrm{n}}^2}{2\mathrm{g}}=\frac{{\left({\mathrm{e}}_{\mathrm{n}}\mathrm{v}\right)}^2}{2\mathrm{g}}=\frac{{\mathrm{e}}^{2\mathrm{n}}{\mathrm{v}}^2}{2\mathrm{g}}$   ... (ii)

Using eqn. (i) in (ii), we get

$\mathrm{h}=\frac{2{\mathrm{g\ell e}}^{2\mathrm{n}}}{2\mathrm{g}}={\mathrm{\ell e}}^{2\mathrm{n}}$                                               ... (iii)

If θ_n is the angle the string subtends wtih vertical after n rebounds,it follows from figure that

h = OB – OD = ℓ – ℓ cos θ_n = (1 – cos θ_n) ...  (iv)

From eq. (iii) and (iv), we have

ℓ e²ⁿ = ℓ (1 – cos θ_n) or e2n = (1 – cos θ_n)

From qn to be less than 60°, i.e. cos qn is greate than $\frac{1}{2}$,

  (1 – cos θ_n) must be less than $\frac{1}{2}$  

  Thus, e²ⁿ < $\frac{1}{2}$

  $\mathrm{Given},\mathrm{e}=\frac{2}{\sqrt{5}}$

  $={\left(\frac{2}{\sqrt{5}}\right)}^{2\mathrm{n}}<\frac{1}{2}\mathrm{or}{\left(\frac{4}{5}\right)}^{\mathrm{n}}<\frac{1}{2}$

or (0.8)n < $\frac{1}{2}$ or  n ln  (0.8) <  ln (0.5)

or – (0.0969) n < – 0.3010 or n > $\frac{0.3010}{0.0969}$

or         n > 3.1

The least integer treater than 3.1 is 4. Hence n = 4. Thus, after 4 impacts,the amplitude will becomes less than 60°.