How we can find the escape velocity from a planet for an object orbiting it?
When we have an object orbiting a given planet, if we assume that the orbit is circular, there must be a force acting on the object, due to the planet, that commits it to follow a circular trajectory. As in any trajectory of this class, this force is a centripetal force, which is equal to m*v^2/R, being R the distance from the object to the center of the planet. Now, at the same time, this force is not a new force, it's the same gravitational force given by Newton's Law. As this force is a conservative one, we can define a gravitational potential energy. Now, if we want to take the object to the infinite from his orbit, leaving it essentially free from the attraction from the planet, we can simply apply the conservation of energy. While it's orbiting at a speed v,, at a distance R from the center of the planet, the total energy is the sum of the kinetic energy and the gravitational potential energy. When at in the infinite, both energies at 0. So, in order to keep the total energy constant, it must be true that the sum of the kinetic energy (at a speed equal to the escape velocity) and the gravitational potential energy must be 0 also. Solving for vescape, we got Vesc= sqrt(2*G*Mp/rp). .
In simple words, can you explain me why current flows in a DC circuit?
No problem, we're here to help. Let's consider a simple DC circuit, with a battery and a resistor linked by ideal wires. In order to understand why current flows, let's think about a gravitational analogy. Let's suppose that you are a skier on the top of the hill, and you want to go downhill. In order to be do so, first of all, there must be a difference in height. In the same way, in order to current flow in a circuit, there must be a difference in "electric height" (that we call potential difference, or simply voltage) that pushes the charges "downhill" turning this potential energy in heat while they go across the resistor. Now, getting back to the skier, once he reaches to the bottom of the hill (ground level), if he wants to go down hill again, he must go to the top of the hill again first. Therefore, there must be an external force that counteracted the gravity, doing work against the force of gravity, taking him to the top of hill again. This force is usually supplied by a skylift. In the case of the current, once the charge carriers passed through the resistor, they are at the same point as the skier at ground level; They need to be taken through the battery by an external force, to the same point where they started to travel "downhill". This force is supplied by chemical energy, which counteracts the electric force pushing the charges downhill inside the battery. In this way, charges are taken again to a "electric height" level, from which they do downhill, and the process repeats continuously, taking the chemical energy from the battery and dissipating it as heat in the resistor.
In simple words, can you explain why a rolling ball that falls from a table follows a parabolic trajectory?
No problem, we're here to help. When the ball is rolling, if there are no external forces acting on it in the horizontal direction, his horizontal velocity is constant. Now, in the vertical direction, while is on the table, as it is not accelerating in this direction, the net force acting on it in this direction must be 0. We have two forces acting in this direction: the normal force, acting upwards, and the force of gravity, acting downwards. When the ball loses contact with the table, the normal force disappears, so it is only subject to the influence of gravity, that always go downwards. As gravity is the only force acting on the ball, and it has no component in the horizontal direction, the net horizontal force is 0, so in this direction, the ball keeps moving at the same speed as it was moving along the table. From the vertical point of view, as it had no speed in this direction when left the table, the ball is just in freefall, like it were starting from rest, and with an constant acceleration equal to -g, so we can apply here the kinematic equations. As we have a constant speed in the horizontal direction, and a downwards increasing speed in the vertical direction, the vectorial sum of both velocities gives as a result a parabolic trajectory, which can be easily proved with just a liitle bit of algebra.