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) .

b. When do the particles collide?

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 of T. and the position of the second point is given by X2 equals 3 cosine of T plus pi divided by 2. When did the two points collide on the interval from 0 to pi? We have 4 possible answers being pi divided by 4, pi divided by 35 divided by 2, and 3 pi divided by 4. Now, to solve this, let's set our positions equations equal to each other. Yeah, X1 equals X2. We need to see where the time is the same for both functions. So let's solve for tea. We have 3, cosine T equals 3. Cosine T plus pi divided by 2. Now, we know, the equation on the right side of the equal sign is a trigonometric sum identity. We'll take our cosine. Of T plus pi divided by 2 and split it up into the sum identity. Cosine, cosine pi divided by 2. Minus sinet spi divided by 2. This gives us cosine T multiplied by 0. Minus sinet multiplied by 1, or just giving us Negative sine T We can now substitute this in our equation. 3 cosine T equals 3 multiplied by negative sine T. If we then divide by 3 on both sides, we get cosine T equals negative sine T. Finally If we divide both sides by negative cosinet, We get 91 equals tangent of T. Now, let's solve for T by getting rid of tangent. To get rid of the tangent, we will take the inverse tangent on both sides. Tangent inverse of -1. Equals tangent inverse of the tangent of T. Or T equals tangent inverse of -1. Now, on the range of tangent inverse, there is one place where this works, and that is when T equals negative pi divided by 4. So the general solution for tea. Will then be Negative pi divided by 4. Plus K pi. Where K is an integer. Now, keep in mind the problem did say we want to find the value on the interval. From 0 To pay. So, let's find values for K to add to our equation. Let's say K equals 0 1st, and then K equals 1, and K equals 2. We have T equals negative pi divided by 4. Plus 0 multiplied by pi. This is just negative pi divided by 4. For K equals 1, we have negative pi divided by 4, plus 1 multiplied. Buy a pie To give us negative pi divided by 4 plus pi. This simplifies to 3 pi divided by 4. And then for 2 We have T equals negative pi divided by 4, plus. To pay Which gives us 7 pi divided by 4. Now, when K equals 0, our value for T is not in the interval, and when K equals 2, the value is also not in the interval. The only value that's in our interval is when T equals 3 pi divided by 4. So that means then, the answer to our problem is T equals 3 pi divided by 4. Which corresponds 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) .

b. When do the particles collide?

Key Concepts

Particle Position Functions

Collision Condition

Trigonometric Identities

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