Explanation
Angular frequency, $$\omega =\sqrt{\dfrac{k}{m}}\ \ \ \Rightarrow \ \dfrac{2\pi }{T}=\sqrt{\dfrac{k}{m}}$$
$$\Rightarrow \ k=m{{\left( \dfrac{2\pi }{T} \right)}^{2}}$$ where, $$T$$ is time period.
Net Spring constant when two spring is connected in parallel.
$$ k={{k}_{1}}+{{k}_{2}} $$
$$\Rightarrow m{{\left( \dfrac{2\pi }{T} \right)}^{2}}=m{{\left( \dfrac{2\pi }{{{T}_{1}}} \right)}^{2}}+m{{\left( \dfrac{2\pi }{{{T}_{2}}} \right)}^{2}}$$
$$ \Rightarrow \dfrac{1}{{{T}^{2}}}=\dfrac{1}{{{T}_{1}}^{2}}+\dfrac{1}{{{T}_{2}}^{2}} $$
Given that,
Amplitude = $$A$$
Time period = $$T$$
Average velocity in time period = $$\dfrac{T}{4}$$
The displacement equation of SHM
$$x=A\sin \omega t$$
We know that,
$$ v=\dfrac{dx}{dt} $$
$$ v=\dfrac{d\left( A\sin \omega t \right)}{dt} $$
$$ v=A\omega \cos \omega t $$
Now, the average velocity is
$$ <v{{>}_{0\to \dfrac{T}{4}}}=\dfrac{\int\limits_{0}^{\dfrac{T}{4}}{vdt}}{\int\limits_{0}^{\dfrac{T}{4}}{dt}} $$
$$ <v>\,=\dfrac{\int\limits_{0}^{\frac{T}{4}}{A\omega \cos \omega t}}{\dfrac{T}{4}} $$
$$ <v>=\dfrac{4A\omega }{T}\int\limits_{0}^{\frac{T}{4}}{\cos \omega tdt} $$
$$ <v>=\dfrac{4A\omega }{T}\left[ \frac{\sin \omega t}{\omega } \right]_{0}^{\dfrac{T}{4}} $$
$$ <v>=\dfrac{4A}{T}\left[ \sin \dfrac{2\pi }{T}\times \dfrac{T}{4}-0 \right] $$
$$ <v>=\dfrac{4A}{T} $$
Hence, the average velocity is $$\dfrac{4A}{T}$$
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