Last time we introduced GaspardGustave de Coriolis’ formula to compute kinetic energy. Today we’ll use it to determine the speed of descent, or velocity, of the coffee mug we’ve been watching closely in the last few blogs. To calculate the mug’s velocity, we must bear in mind physicist Julius Robert von Mayer’s assertion that all forms of energy are interrelated, and in fact interchangeable, because energy can neither be created nor destroyed, it can only change forms. For a refresher, see The Law of Conservation of Energy. Let’s now put a practical spin on this concept and apply it to our coffee mug’s free fall to the floor. Once again, de Coriolis’ formula, KE = ½ × m × v^{2 } (1) where m is the mass of our falling object and v its velocity. The ½ is an unchanging, constant term that’s present due to the mathematical Rules of Integration governing integral calculus. Calculus and its derivations are beyond the scope of this blog, but if you’re interested in pursuing this, follow this link to, The Physics Hypertextbook – Kinetic Energy. According to von Mayer’s Law, at the precise instant before the mug hits the floor its kinetic energy, KE, is equal to the potential energy, PE, it possessed when it rested passively on the shelf. Stated another way, the instant before the mug makes contact with the floor, all its potential energy will have been converted into kinetic. The mug’s PE was calculated previously to be equal to 4.9 kg • meter^{2}/second^{2}. See Computing Potential Energy for a review. Knowing this, the mathematical relationship between the mug’s potential and kinetic energies is expressed as, PE = KE = 4.9 kg • meter^{2}/second^{2} (2) By substituting this mathematical representation for KE into equation (1) we arrive at, 4.9 kg • meter^{2}/second^{2 } = ½ × m × v^{2} (3) We also know the mug’s mass, m, to be equal to 2.6 kilograms, so integrating that into the right side of equation (3) it becomes, 4.9 kg • meter^{2}/second^{2 } = ½ × ( 0.25kg) × v^{2} (4) That leaves the mug’s velocity, v^{2}, as the only remaining unknown term. We’ll use algebra to isolate this variable by dividing both sides of equation (4) by ½ × ( 0.25kg). (4.9 kg • meter^{2}/second^{2}) ÷ [½ × ( 0.25kg)] = v^{2} 39.20 meter^{2}/second^{2} ^{ }= v^{2} Finally, we’ll take the square root of the equation to place it in terms of v. 6.26 meters/second ^{ }= v The mug’s velocity an instant before impact equates to 6.26 meters/second, or almost 21 feet per second. Next time we’ll discuss a metric unit used to measure energy known as the Joule and discover the man behind it. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________ 
Posts Tagged ‘Julius Robert von Mayer’
Calculating Velocity — de Coriolis’ Kinetic Energy Formula
Monday, September 28th, 2015Willem Gravesande’s Experimentation on Kinetic Energy
Friday, September 11th, 2015
Last time we introduced The Law of Conservation of Energy, which holds that energy can neither be created nor destroyed. We then applied the concept to a mug resting on a shelf, brimming with latent gravitational potential energy. Today we’ll continue our discussion with a focus on kinetic energy and how Willem Gravesande’s experimentation contributed to our understanding of the subject. The concept of kinetic energy was first posited by mathematicians Gottfried Leibniz and Johann Bernoulli in the early 18^{th} Century when they theorized that the energy of a moving object is a factor of its mass and speed. Their theory was later proven by Willem Gravesande, a Dutch lawyer, philosopher, and scientist. Gravesande conducted experiments in which he dropped identical brass balls into a soft clay block. See Figure 1. Figure 1 Figure 1 shows the results obtained when balls of the same mass m are dropped from various heights, resulting in different velocities as they fall and different clay penetrations. The ball on the left falls at velocity v and penetrates to a depth d. The center ball falls at twice the left ball’s velocity, or 2v, and penetrates four times as deep, or 4d. The right ball falls at three times the left ball’s velocity, 3v, and it penetrates nine times deeper, 9d. The results indicate an exponential increase in clay penetration, dependent on the balls’ speed of travel. In fact, all the kinetic energy that the balls exhibited during freefall was converted into mechanical energy from the instant they impacted the clay until their movement within it stopped. This change in forms of energy from kinetic to mechanical demonstrates what Julius Robert von Mayer had in mind when he derived his Law of Conservation of Energy. For a refresher on the subject, see last week’s blog, The Law of Conservation of Energy. As a result of his experimentation, Gravesande was able to conclude that the kinetic energy of all falling objects is a factor of their mass multiplied by their velocity squared, or m × v^{2}. We’ll see next time how Gravesande’s work paved the way for later scientists to devise the actual formula used to calculate kinetic energy and then we’ll apply it all to our coffee mug falling from the shelf. Copyright 2015 – Philip J. O’Keefe, PE Engineering Expert Witness Blog ____________________________________
