Particle motion At time t, the position of a ...

Question

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.

c. Find the total distance traveled by the body from t = 0 to t = 2.

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A particle is moving along a line such that its position at time T in seconds is given by PF T is equal to parentheses 23 minus 92 plus 12 centimeters. Find the total distance the particle travels from T equals 0 to T equals 3. Awesome. So it appears for this particular problem we're asked to use our function that is provided to us to help us determine the total distance the particles travel from T equals 0 to T equals 3. So ultimately we're trying to figure out what the total distance value is that this particle travels during the time period from 0 seconds to 3 seconds. So with that said, let's read off our multiple choice answers to see what our final answer might be for the total distance, noting that they're all in the same units of centimeters. So A is 19, B is 21, C is 9, and D is 11. Awesome. So first off, our first step that we need to take to solve this particular problem is we need to compute the velocity function. So in order to compute the velocity function, we need to recall and note that velocity is the derivative of the position. So let us note that V of T. Is going to be equal to D by DT of 23 minus 9 T 2 + 12T. Fantastic. So when we perform the derivative, that will mean that VFT is going to be equal to 6 T 2 minus 18T plus 12. Fantastic. So now our second step is we need to find where the velocity is zero. So as we should recall a note, a change in the direction occurs when V of T is equal to 0. So that means we need to take 6 T 2 minus 18 T + 12 and set it equal to 0. So now what we need to do is we're trying to ultimately rearrange this equation to isolate and solve for T. So I'm gonna skip a few steps in an effort to save time solving, but you start by dividing everything by 6, and then you factor, and when you factor, you should get T minus 1 multiplied by T minus 2 is equal to 0, so that will mean that T equals 1 and T equals 2. So these values are within our interval, square bracket 0.3 square bracket square bracket where the square brackets indicate a closed interval. So that means that the particle changes direction at T equals 1 and T equals 2. So they'll change direction at T equals 1 and T equals 2. So with that said, our third step now is we need to compute the position at T equals 0.1.2.3. Awesome. So right now, our next step now is once again, we're trying to compute the position at T equals 012, and 3. So with that said, Let us take that piece of information and what we're gonna be doing, since once again we're computing the position at the specific time, so we need to take, let's start with 0, P is 0, so we're plugging in a value of 0 in for T. So that will mean that P of 0 is equal to 2, multiplied by 0 to the power of 3 minus 9 multiplied by 0 to the power of 2 plus 12 multiplied by 0, which is going to be equal to 0. So in an effort to save time, instead of writing out every single equation for each of these, I'm just gonna write what P of 12, and 3 is. So P of 1. TF 2 and P3. So I'm just, so essentially we're just plugging in different values for T into our original equation, or rather our original function. So in an effort to sit down, I'm not going to write each each equation out over and over again. I'll just give you the evaluated answer. So P1 is going to be equal to 5, PF 2 is going to be equal to 4, and P of 3 is gonna be equal to 9. So now, We need to for our 4th step. So for step number 4, we need to compute the total distance traveled. So as we should recall and note, the total distance, as we should recall, will be the sum of the absolute position changes. So that will mean that the distance D will be equal to the absolute value of P 1 minus P 0. Plus, the absolute value of P 2 minus 1. Plus, the absolute value of P 3 minus P2, fantastic, and don't forget that this is also an absolute value sign. So that will mean when we plug in all of our known variables, this will be equal to the absolute value of 5 minus 0, plus the absolute value of 4 minus 5. Plus, the absolute value of 9 minus 4. So when we plug All of this information into our calculator. So if we take 5 minus 0, we'll give us 5, and then 4 minus 5 is -1, but it's an absolute value sign, so it's going to be 1. And then 9 minus 4 is going to be 5, so plus 5, so 5 + 1 plus 5 is going to give us an answer of 11 centimeters. So that means without a doubt, our final answer has to be 11 centimeters. So to quickly recap, the total distance traveled by the particle from T equals 0 to T equals 3 will be 11 centimeters, and this corresponds with multiple choice answer. Letter D, which once again is 11 centimeters. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.

c. Find the total distance traveled by the body from t = 0 to t = 2.

Key Concepts

Derivative

Critical Points

Total Distance Traveled

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