Why does velocity of electron increases with increase atomic number in the Bohr model of the hydrogen and hydrogen like atoms? I already know mathematical proof which states velocity of electron increases with increase in atomic number,but what is the intuition behind it?  
 A: The same reason the Earth would move faster should the mass of the sun increase: centripetal force grows. In the case of Bohr's model, the force grows as $\sim Z$, 
$$
F_e = \frac{k Ze^2}{r^2}
$$
Newton's law thus results in 
$$
\frac{mv^2}{r} = F_e = \frac{k Ze^2}{r^2}
$$
leading to $v \sim Z^{1/2}$
A: As others have mentioned already, it is a classical mechanics problem. Assuming an already circular orbit, the magnitude of the centripetal force (v^2/rho) is what is necessary to keep the object in circular orbit at speed v.  This force is generated by the nucleus, and this relationship is bilateral, meaning that if the force is applied already (again assuming an already circular orbit), this will speed up the electron until it meets the condition.
This is a good illustration of how centripetal acceleration works. If the relationship Fc=v*2/rho holds, then this acceleration is only directional and will keep the object in orbit. However, if we have an object in circular orbit at speed v, and Fc increases, then there will be a tangential acceleration even though we have just increased the magnitude of the acceleration in the normal direction, not the tangential direction. The tangential acceleration will act only until the new v is acquired to satisfy the relationship, then it will cease, hence it is an momentary impulse rather than a constant acceleration, and cannot be included when laying out Newton's equations for a system. They assume a static configuration, in that all accelerations are constant, there cannot be any momentary impulses.
This is why students have such a hard time grappling things like the gyroscopic effect, which comes down to the same. Consider a spinning wheel, which is radially parallel to a string from the ceiling attached to its central axis. Gravity acts on the wheel, but when it acts on the top part, this top part is already at the side (because it is spinning), and it gravity moves the whole thing sideways. What is happening here is that the second you let the wheel go (and let gravity act), the centripetal force of the wheel-around-string motion increases (because gravity acts parallel to the floor in this case, not perpendicular to it as is normal, it acts on all sides, but the sum is inwards towards the center), and for only a second there is an impulse which equals the orbit velocity to what is dictated by the constant centripetal force. This impulse cannot be included in Newton's equations, it is not a static situation. We can however compute the situation when everything is moving with constant acceleration. The phenomenon of the gyroscopic effect itself happens at the impulse though, which is why students struggle to understand it.
A: As  the electron moves upward force of attraction betn the nucleus is decreased which result into decreased in acceleration as the acceleration decreased velocity of the electron decreased
A: As the electron moves farther away from the nucleus, the potential energy of the electron decreases [ the electrostatic potential between the nucleus and the electron decreases with distance]. The kinetic energy has to increase, to keep the total energy constant. This directly implies an increase in velocity
A: The total energy if the energy of the electron in the Bohr atom must be constant, since no agent is doing work on the electron. As the electron moves farther away from the nucleus, the potential energy of the electron decreases [ the electrostatic potential between the nucleus and the electron decreases with distance]. The kinetic energy has to increase, to keep the total energy constant. This directly implies an increase in velocity.
