We wondered last week how the formula for the kinetic energy of a body of mass m is derived at speed v: Ec = mv² / 2. Once again, we will resort to the law of conservation of energy, since the kinetic energy of the body in question must be equal to the work necessary to print its speed.
Let us consider the case of a body that falls during t seconds: the work (W) done by the gravitational attraction is equal to the force – which is the weight of the body (mass by gravity: mg) – through the space traveled, which in the case of a body in free fall is gt² / 2; therefore, and since gt = v (the velocity is the acceleration of gravity times the fall time):
Ec = W = F.e = mg.gt²/2 = mg²t²/2 = mv²/2
In order to refresh some notions of elementary physics, a body in free fall for t seconds starts from rest, that is, speed 0, and reaches the speed gt, so its average speed in those t seconds is gt / 2, and multiplying the average velocity over time we obtain the space traveled: gt² / 2, which, as we have just seen, gives us the formula for kinetic energy when multiplied by the force, mg.
In view of the above considerations, a “naive” observer (unfamiliar with relativity) might think that the famous formula for the equivalence between matter and energy, E = mc², expresses the kinetic energy of a body of mass m that will instantaneously reach the speed of light (hence the disappearance of the factor ½, since it would not start from rest or speed 0). I invite my astute readers to reflect on it.
The mechanical equivalent of heat
After contemplating two extreme forms of energy conversion: the “classical” conversion of potential energy into kinetics and the relativistic conversion of matter into energy, it is necessary to mention the conversion of work into heat (and vice versa), a concept not as revolutionary as the one introduced by Einstein, but which science was not clear about until the nineteenth century.
It was not even clear that heat was a form of energy, as it was thought to be a kind of subtle fluid (called “caloric”) that permeated bodies and passed from one to another. Despite the numerous evidences that mechanical work can produce heat (for example, by rubbing an object), this relationship was only seen clearly and could be quantified from the experiments carried out by the British physicist James Prescott Joule in the middle 19th century.
Joule determined that to raise the temperature of a gram of water by one degree, that is, to generate a calorie, it was necessary to use slightly more than four joules of mechanical energy
In a container of water, Joule introduced rotating paddles connected by a rope to a weight that, as it fell, rotated the paddles, converting the potential energy of the weight into mechanical energy (the rotation of the paddles) which in turn made the water temperature rise; that is, mechanical energy was transformed into heat.
With this type of experiment, Joule determined that to raise the temperature of a gram of water by one degree, that is, to generate a calorie, it was necessary to use a little more than four joules of mechanical energy. Subsequently, it was accurately determined that the equivalence between units of heat and energy is 1 cal = 4.18 joules. Recall that one joule is the work done by a force of one newton when traveling a space of one meter (approximately the work necessary to lift a weight of 100 grams to one meter in height).
Are we intoxicated with misleading advertising or are there really stoves that use less than others?
And speaking of heat, which we have had plenty of this summer, will soon go away and return, with the cold, the advertising of all kinds of electric stoves, which often boast of their low consumption. Do they poison us, in doing so, with misleading advertising, or are there really stoves that use less than others? And what about six-watt bulbs that shine like 40-watt bulbs? What about high-performance refrigerators? And the rigorous meta-question: what does all this have to do with energy conversion and conservation?
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.
Eddie is an Australian news reporter with over 9 years in the industry and has published on Forbes and tech crunch.