Particle motion The positions of two particle...

Question

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

a. What is the farthest apart the particles ever get?

Accepted Answer

Two points are moving along the x-axis. The position of the first point at any time T is given by X1 equals 3 cosine T, and the position of the second point is given by X2 equals 3 cosine of T plus pi divided by 2. What is the maximum distance between the two points on 0 to 5 divided by 2? We have 4 possible answers, being squared of 232, or 3 squares of 2 pollen units. Now, to stop this, I'll first write out our equations, 3 cosine T. And 3, cosine T plus pi divided by 2. Now, we notice that we can rewrite X2 using Prayed some formulas. Our tricks some identity for Kosun. will be Per Of T multiplied by cosine, I divided by 2. Minus sine of T multiplied by a sign of pi divided by 2. We get 3. Multiplied By cosine T multiplied by 0. Minus T. Multiplied by one. Which gives us -3, sin T. Now, the distance, will be the absolute value. X1 minus X2. Which is the absolute value. Of 3 cosine T. Minus -3 st. Which further simplifies to the absolute value. Of 3 cosine T plus 3, sin a T. Now this expression is always non-negative on our interval. We can then just confirm that the absolute value bars disappear, giving us 3 cosinet. Plus 3 T. Now to find our maximum distance, we will take the first derivative. The prime is equals. 2 -3 sin T. Plus 3 cosinet. And now, to maximize, we will set our derivative equals to zero. 0 equals -3 T. Plus 3 cosinet. We can add over our 3 sinet to get 3 st equals 3 cosine of T. We can divide Oh son And we can divide 3 on both sides. We now have sine divided by cosine, which gives us tangent T equals 1. No tangent If we want to sell for tea, We can take the tangent inverse to get rid of that tangent. T equals inverse tangent. Of what On the unit circle, this corresponds with the angle pi divided by 4. No Let's check If this is a maximum. We want to see if our second derivative. It's less than. It's here. This will tell us if this is a maximum. In particular, we want the 2nd derivative at pi divided by 4. So let's take the second derivative. A second derivative Will be -3, cosine T. 3 synopsy. If we plug in, I divide it by 4. We end up getting 3 squares of 2. This value is less than 0, so we can confirm that this is the maximum. Now, to determine the maximum distance, We are going to plug in our values. So, We have D of pi divided by 4. And we can also check D of 0. And D of pi divided by 2. If we plug in our face. We get 3, plus sign, I divided by 4. Plus 3 sign pi divided by 4. We end up getting the value 3 square roots of 2. If we plug in the 0, we get 3, and if we plug in pi divided by 2, we also get 3. All in units. We can then confirm The max distance This equals 3 squares of 2. Units If we look at our possible answers, this would correspond. To answer D. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Particle motion The positions of two particles on the s-axis are s₁ = cos t and s₂ = cos (t + π/4) .

a. What is the farthest apart the particles ever get?

Key Concepts

Position Functions

Distance Between Two Points

Maximization Techniques

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