You are correct, and I was wrong in the amount of energy necessary to bring the water to a boil. I was using wolfram alpha, "energy to boil a cup of water", which probably then gave me the energy needed to boil it all away, not just to bring it to a boil.
It takes a tremendous amount of energy to heat things, overall. For example, compare the energy in a bullet to the energy in a warm cup of water. A 20g bullet moving at 400 m/s has 1600 joules of energy. That amount of energy, when used to warm a cup of water, would only warm it by less than 2 degrees Celsius.
Let's start with your last statement, on the relationship between energy, momentum, and mass. The most common formula is E=mv^2/2. This works for objects with mass that are moving slowly. Now, by "moving slowly", I mean "relative to the speed of light". The fastest object made by man, the Juno spacecraft, moves at about 140,000 km/hr, which is a paltry 0.00013c. This formula still works here.
If we want to know what happens at faster speeds, we need to dip into special relativity. There are some messy derivations, but one critical formula that comes out of it is E=sqrt(m^2c^4 + p^2c^2). If you imagine a ninety-degree triangle where the shorter sides are the mass of an object and the momentum of an object, then the long edge is the energy. Increasing either mass or momentum will increase the energy.
Suppose we start dialing down the momentum in this equation. After some math, (http://en.wikipedia.org/wiki/Binomial_theorem ), we arrive at E=mc^2 + mv^2/2. This is the origin of the famous E=mc^2. We now have two terms, the rest energy, and the kinetic energy. Note that the other term is the more familiar form of kinetic energy, which is why it works in everyday life.
Now, suppose we go the other route and set m=0. Then we get E=pc. Even though one side of the triangle (mass) has been reduced to zero, the other side (momentum) still gives it a non-zero energy. I apologize if that is more math than you had been hoping for, but eventually one always runs into some math.
A few things on the spaceship. You are absolutely right on the thinking of it. As soon as the laser switches on, it starts applying force. If my friend's spacecraft is one light-second away, I don't need to wait one second for the laser to reach her before feeling anything, because the light itself carries momentum away. In fact, if I wanted, I could just fire the laser out into space. I get propelled one way, and to balance out the momentum, the photons are travelling in the opposite direction.
It takes a tremendous amount of energy to heat things, overall. For example, compare the energy in a bullet to the energy in a warm cup of water. A 20g bullet moving at 400 m/s has 1600 joules of energy. That amount of energy, when used to warm a cup of water, would only warm it by less than 2 degrees Celsius.
Let's start with your last statement, on the relationship between energy, momentum, and mass. The most common formula is E=mv^2/2. This works for objects with mass that are moving slowly. Now, by "moving slowly", I mean "relative to the speed of light". The fastest object made by man, the Juno spacecraft, moves at about 140,000 km/hr, which is a paltry 0.00013c. This formula still works here.
If we want to know what happens at faster speeds, we need to dip into special relativity. There are some messy derivations, but one critical formula that comes out of it is E=sqrt(m^2c^4 + p^2c^2). If you imagine a ninety-degree triangle where the shorter sides are the mass of an object and the momentum of an object, then the long edge is the energy. Increasing either mass or momentum will increase the energy.
Suppose we start dialing down the momentum in this equation. After some math, (http://en.wikipedia.org/wiki/Binomial_theorem ), we arrive at E=mc^2 + mv^2/2. This is the origin of the famous E=mc^2. We now have two terms, the rest energy, and the kinetic energy. Note that the other term is the more familiar form of kinetic energy, which is why it works in everyday life.
Now, suppose we go the other route and set m=0. Then we get E=pc. Even though one side of the triangle (mass) has been reduced to zero, the other side (momentum) still gives it a non-zero energy. I apologize if that is more math than you had been hoping for, but eventually one always runs into some math.
A few things on the spaceship. You are absolutely right on the thinking of it. As soon as the laser switches on, it starts applying force. If my friend's spacecraft is one light-second away, I don't need to wait one second for the laser to reach her before feeling anything, because the light itself carries momentum away. In fact, if I wanted, I could just fire the laser out into space. I get propelled one way, and to balance out the momentum, the photons are travelling in the opposite direction.