Lecture Note
University
Rice UniversityCourse
Preparing for the AP Calculus AB ExamPages
2
Academic year
2023
Awayne
Views
49
Particle Motion ● Particle motion is the study of how particles move. It is a branch of physics and chemistry, which deals with the motion of particles at the atomic or molecular scale. ● The study of particle motion is important in many fields of science, including physics, chemistry, biology and geology. Understanding the motion of particles is basic tounderstanding the world around us. ● Particle motion describes the movement of small objects like atoms and molecules under the influence of external forces such as electric and magnetic fields or gravity. We'll tie your calculus work to the subject of particle motion, which you've already studied inphysics. When mathematicians were trying to solve the tangent line problem, they also wanted to explorehow it related to other questions they were working on, such as the relationship between velocityand acceleration. As it turns out, calculus and physics are closely related disciplines.
We have the following equation: x of t equals x of 0 plus 2t. The average velocity on the intervalfrom zero to two is the change in position, which is x of 2 minus x of 0, divided by the change intime, which is 2 minus 0. Let's find those two values. Let's see here. 8 minus 16 plus 6 minus 1, so negative 8. Negative 9, so negative 3; and then plugin 0, you get negative 1. So if we consider the equation negative 3 minus 1, then we notice that the average rate of changeis negative 1. Yes, the average velocity of the particle is negative 1, but we note that it is identicalto slope, which we know to be an average rate of change. We effectively found the average rate of change of position by using the equation slope of positionequals velocity. Using the equation rate of change equals slope, we can see that velocity is the derivative ofposition. The derivative of velocity gives you position. Velocity, which can be thought of as a rate of changein position, is itself a velocity. But what if we considered the slope or rate of change of velocity? Your change in velocity over change in time. Let's try units. Velocity is meters per second. Time isseconds. Now we have meters per second squared. As you know, the metric measurement of acceleration is meters per second squared. Then the rateof change of velocity is acceleration. And surely, I can interchange the word "rate of change" with "derivative." So the derivative ofvelocity is acceleration. And so that's just some basic math you have to know. So long as the variable that changes its position in the function is indicated by a different letter, youcan use any letter you like to represent the dependent variable. x of t, y of t, s of t, p of t. As long as the particle stays in the same position, its velocity is constant. The acceleration of aparticle is directly proportional to the derivative of its velocity. Thus, if you differentiate thederivative of position with respect to time, you get acceleration. So the second derivative of position, which we denote as the derivative with respect to time, isacceleration. Next, I'll show you how to find the acceleration of an object by taking the secondderivative of its position. You may recall from your physics class that this is a kinematic equation for position. The position ofan object at any time t can be calculated by the following equation: Initial Position + Velocity TimesTime + 1/2 Acceleration Times Time Squared. For a more formal definition, we can take the derivative of x0, which is a constant. Since thederivative of a constant is 0, we can say that x0 is a constant. We're taking the derivative with respect to time, so the derivative of vt is v. It's probably some sortof v0 and a0. The derivative of 1/2 a t squared is not a variable. The product rule applies only to variables, notconstants. We are left with the kinematic equation for velocity, acceleration, and time. And so, just to continue the mathematical trend here, we'll take the derivative of this. Now that'll be equal to v prime of t, which is equal to a of t. The derivative of v0 is equal to 0. Thederivative of a t is equal to a. And what do we see? Acceleration is equal to acceleration. It's almost like the laws of physics werecreated by someone who knew what they were doing.
Particle Motion
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