Horizontal Motion The position function of a particle moving...

Horizontal Motion The position function of a particle moving along the $x$ -axis is $x(t)=A e^{k t}+B e^{-k t},$ where $A, B,$ and $k$ are positive constants. (a) During what times $t$ is the particle closest to the origin? (b) Show that the acceleration of the particle is proportional to the position of the particle. What is the constant of proportionality?

Horizontal Motion The position function of a particle moving along the $x$ -axis is $x(t)=A e^{k t}+B e^{-k t},$ where $A, B,$ and $k$ are positive constants.
(a) During what times $t$ is the particle closest to the origin?
(b) Show that the acceleration of the particle is proportional to the position of the particle. What is the constant of proportionality?

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form f(t) = Ce^(kt), where C and k are constants and t is the variable. They are important in modeling growth and decay, and their properties are often exploited in solving differential equations and in analyzing time-dependent behavior in physical systems.

Minimization Techniques Using Differentiation

Minimization techniques involve finding the points at which a function reaches its smallest value. This is commonly done by taking the derivative of the function with respect to the variable of interest, setting the derivative equal to zero to find critical points, and then using tests such as the second derivative test to determine if these points represent minima.

Acceleration as the Second Derivative of Position

In kinematics, acceleration is defined as the second derivative of the position function with respect to time. This concept is fundamental because it connects the rate of change of velocity to the forces acting on a system. When the acceleration is found to be proportional to the position, it indicates a linear relationship that is characteristic of simple harmonic motion or similar dynamic systems, with the constant of proportionality describing the stiffness or reactive force coefficient.

Transcript

00:01 Okay, what we're going to do is we're going to talk about horizontal motion.

00:04 And so we're given a position function as x of t is equal to a, e to the kt, plus b, e to the negative kt.

00:11 Where a, b, and k are just positive constants.

00:15 And the first thing we want to do is we want to determine the time when the position of the particle or whatever is moving is closest.

00:33 To the origin.

00:36 Okay, so if we think about this, we want to minimize that position.

00:42 We want the minimal position.

00:43 And so we have to think, okay, maximums and minimums occur at our critical numbers, where our first derivative actually equals zero.

00:54 And so that's, we're going to take the first derivative.

00:58 And so this would be a, k, e to the kt, minus bk, e to the negative kt.

01:07 Okay, and so we're going to go ahead and put this over a common denominator.

01:12 So we have ak, oops, e to the 2kt minus bk all over e to the kt.

01:28 Okay, and so we're going to set this equal to zero because we're solving for the critical numbers where that derivative equals zero.

01:36 And of course, it's going to be where your numerator equals zero.

01:45 And we're going to solve for t.

01:47 And so when we do that, we get e to the 2kt is equal to b over a.

01:55 And so, of course, we're going to take the natural log of both sides.

01:59 And so we get t is equal to 1 over 2k natural log of b over.

02:08 So there is our critical number...

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