Give the position vectors of particles moving along various curves in the x y -plane. In each ca

Give the position vectors of particles moving along various...

Key Concepts

Chain Rule in Differentiation

The chain rule is a key tool in calculus used for differentiating composite functions. When the parametric equations for a curve involve functions like cosine and sine of a function of time, the chain rule is applied to correctly compute the derivatives needed to find the velocity and acceleration vectors.

Acceleration Vector

The acceleration vector is the time derivative of the velocity vector. It measures the rate of change of the velocity, thereby encapsulating changes in the speed or direction of motion. In many cases, especially in circular motion, the acceleration vector can be decomposed into components that are tangential and normal to the curve.

Circular Motion

Circular motion involves a particle moving along the circumference of a circle, which can be described using sine and cosine functions in its parametric form. The constant distance (radius) from the center simplifies analysis, as the velocity vector is always tangent to the circle and the acceleration vector often points radially inward (centripetal acceleration).

Parametric Representation of Curves

A parametric representation expresses the coordinates of a moving particle as functions of an independent parameter, typically time. This method allows one to describe complex trajectories in the plane or space in a clear and concise manner, making it especially useful for analyzing motion along curves.

Velocity Vector

The velocity vector is the derivative of the position vector with respect to time. It describes both the speed and the instantaneous direction of the moving particle, and for motion along curves, the velocity is tangent to the path at each point.

Transcript

0:00 Hi there.

00:01 In this problem, we're asked first to find the velocity and acceleration vectors for these two times over here, so pi and three pi over two.

00:13 Let's begin by remembering that velocity is just the derivative of position, so we need to find r prime of t.

00:23 To do that, we can just take the derivative of each component of r separately.

00:28 So i'll use angle brackets here instead of the i and j notation, but feel free to use whichever you prefer.

00:34 The derivative of 4 cosine t over 2.

00:39 Now the derivative of cosine is minus sign.

00:41 So the 4 is a constant that will stay in there.

00:45 We'll end up with minus sign.

00:49 We keep the input t over 2 and now by the chain rule we need to multiply by the derivative of t over 2 which is 1 half.

00:59 So that's our i component and our j component likewise we'll have a 4 derivative of sine is cosine and keep the input but then multiply yin because of the chain rule by a half.

01:14 And that's velocity, so let's clean that up just a little bit.

01:20 Negative four times a half, of course, is minus two.

01:24 Everything else is the same.

01:26 And in the second component, again, four times a half gives us two.

01:32 And so there's our velocity vector for any time t.

01:36 To get acceleration, acceleration is just the derivative of velocity, which we just found just now.

01:48 And so looking at our answer here for velocity, let's take each derivative of each component separately.

01:56 The derivative of minus 2 sine t over 2.

01:58 Well, the minus will stay there.

02:00 The minus 2 stays there.

02:02 The derivative of sine is cosine.

02:04 We keep the t over 2.

02:06 And then once again by the chain rule times a half.

02:09 And in the second component, we keep the 2.

02:12 The derivative of cosine is minus.

02:16 Sine and again by the chain rule times a half.

02:20 So simplifying negative two times a half we'll just have minus cosine t over two in the first component and let's see minus sine t over two in the second component.

02:34 Okay so that's how i get velocity and acceleration at any time t.

02:40 Now we specifically want these certain times pi and three pi over two.

02:44 So let's begin with t equals pi so on time equals pi let's get velocity first of all the velocity let's look up here this is our velocity vector and this plug in pi for t and see what we get so when we plug in pi for t we'll have pi over two as all of our inputs for our trig here so sign of pi over two let's recall is one and so negative two times one is just negative 2.

03:26 Over in the second component, cosine of pi over 2 is 0.

03:30 So we'll get the 0 in that slot.

03:33 Now i get acceleration at times time pi.

03:36 We just again plug pi into acceleration, which is down here.

03:43 Cosign of pi over 2 we said is 0.

03:46 Sign of pi over 2 is 1.

03:48 So we get minus 1 there.

03:51 Okay.

03:54 And before we go ahead move on to the second time we care about, we're asked to draw these vectors.

04:01 So why don't we do that now? so the curve we're given is right up here x squared plus y squared equal 16.

04:14 So we should recognize that as a circle centered at the radius with the radius, i'm sorry, centered at the origin, with radius the square of 16 or 4.

04:28 So that's our path.

04:31 When time equals pi, this is plug pi into r...

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