Many students have heard about motion at constant acceleration. Unfortunately, pedagogical experiments reveal that what students learned about motion at constant acceleration is wrong, or at least did not mean what they thought it meant. The linked complete chapter deals with this issue. As a sample:
At the start of the book, I supplied the fundamental equation
PHYSICS – CALCULUS = NONSENSE
You will now see this equation in action.
Calculus is not a new tool. It was developed more or less independently by Isaac Newton and Gottfried Wilhelm Leibniz nearly four centuries ago. Newton and Leibniz had a vigorous and somewhat pointless dispute as to who had developed what first. Recently, it became apparent that at about the same time an Austrian monk had also developed the Fundamental Theorem of Calculus. He tragically died, saving a young boy from drowning in a mountain stream, before he could publish his results. It is a curious fact that one of the allowed occupations of Samurai warriors in the Tokugawa period of Japan was abstract mathematics, a skill believed to be as useful as calligraphy or flower arranging. Some historians have made a case that this mathematical school independently developed, using a very different representation of mathematics, the basic ideas of calculus. What you should now have studied is Newtonian calculus, using the more sophisticated notation of Leibniz. As has been said before, Newton was a truly brilliant man, so he didn’t worry whether or not his notation was easy to use or prone to introducing errors. Leibniz viewed himself as writing for mere mortals, and therefore thought carefully about how to make his notation easy to understand and unambiguous in employment.
In this chapter, I provide a short refresher on calculus and show a single application, sometimes described as kinematics or as motion at constant acceleration. As emphasized in the Introduction, this course assumes that you’ve already had enough calculus to be familiar with the integrals and derivatives of standard functions.
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There is also a major point which you have surely all seen but whose significance was not always made apparent. That’s the constant of integration. For example, if I integrate ax with respect to x, taking the indefinite integral in which the limits of integration are not specified, I obtain
\integral ax dx = ax^2/2 + x_0
and not
\integral ax dx = ax^2/2
x_0 is a constant, the constant of integration. Your calculus preparation may not have stressed why constants of integration are important. Later in the chapter, I will show why constants of integration can be critically important, what they do, and how to determine their values.
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We advance to mythical California of 70 years ago and the quaint local custom of drag racing. Two cars driven by folks of limited good sense and even more limited regard for the law pull up at a red traffic light and stop. The character in the arrest-me-red sports car shouts at the character driving the apparent period SUV Hey, man, want to drag?, this being an invitation to race the moment the light changes. When the light changes, the sports car takes off down the highway. The other vehicle seems to sit there. However, the other vehicle is not a period SUV. It is a highly modified vehicle, a Volksraketenwagon (People’s Rocket Wagon), the modification being to replace everything behind the driver with a large liquid-fuel rocket engine hidden by the car’s outer hull. The rocket takes ten seconds to power up, but then more or less instantly delivers a considerable constant thrust, to be precise, enough thrust that the Volksraketenwagon accelerates at A = 100 m/s^2. (That’s ten gravities.) What are the motions of the Volksraketenwagon?
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x(t) = x_0 + v_0 t + 0.5 A t^2.
But what is x_0 ? I have actually had a very few students get this one right when first asked, but very few indeed.
For more, read the chapter.