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How long will it take the particle t0 reach Q?(Use decimal notation. Give your answer t0 (wo decimal places )Entur nlmancuuLa...

Question

How long will it take the particle t0 reach Q?(Use decimal notation. Give your answer t0 (wo decimal places )Entur nlmancuuLa

How long will it take the particle t0 reach Q? (Use decimal notation. Give your answer t0 (wo decimal places ) Entur nlmancuuLa



Answers

A particle that moves along a straight line has velocity $ v(t) = t^2 e^{-t} $ meters per second after $ t $ seconds. How far will it travel during the first $ t $ seconds?

Okay, so, yeah, given quantity Q, which is equal to two point 360 tend to be power minus food, K g Meter square forever. Okay, on the first part is that I have to write Mhm que dash. Oh, in terms off Q with units L B feet. Thank you. Square for a second. Okay, so ours have to be changed two seconds. So how I will write this is Que da's will be cool. Toa que Now I will change all of the's. Okay, So k g would be converted into Yeah, pounds, which is two point 2046 elderly's for K G. And then meter square would be converted into feet. So 1 m is equal to 3.2. Wait, zero it feet squared because this is squared and this is divided by hours, So this will be the other way round, so we'll have one are over 3600 seconds. Okay, so this is how you do it. Next is estimate Que dash approximately. Okay. And then calculated to correct significant figures. Okay, so let's tow the estimation. Wow, This is by rounding off numbers. Do indeed years. Okay, So que dash will be good to queue. So this would be two point 2046 Herbie. So this will be equal to 2.2046 I'm just writing the same equation again. Do it to healthy M or Katie. Three feet. Cool am squared. So calculating these values, this Q is already given to us, right? This value of approximately I have rounded it off on all of the's air here. What I get is three into 10 to the power minus seven. Okay. And now we'll move on to the actual part. Would say is calculate que dash. Okay, let's use a different van Cura. She is equal. Do now. I will substitute the exact values in the situation. Okay, so cue is equal. Do 2.360 10 to the power minus four k g meter each. And then this is 2.204 six every one G. This is 3.280 It feet 1 m square, Etch over 3600 seconds. So, using a calculator, I solved this and I got 4.7415 10 to the power minus seven. Okay? And If I want my answer in, let's say four significant figure Que dash is equal to 4.742 10 to the power minus seven. So this is going to be there.

Section six. Not 1 43 They give us a particle that moved along a straight line has a velocity V of t is t squared you to the minus T meters per second after t seconds. How far will it travel in the first tee second? Well, we know that the velocity is the change in distance with respect to time. Okay. So to find the distance, you know, the velocity is the first derivative I would need to integrate to find the distance of what they're asking me to find is I need to be able to integrate t squared e to the minus T d t. Well, that's gonna be integration by parts. Let u equal t squared. The U is equal to two. T DT d V is equal to e to the minus t t t and then V is equal to minus E to the minus t. So the inter role t squared e to the minus T t t is equal to UV, so minus t squared e to the minus t minus the integral off the d u. And now we repeat that same process again to integrate by parts. The second the integral that we see here letting you equal t do you is equal to d T devi is equal to e to the minus T t t. Therefore, v is equal to minus E to the minus t. So this gives us minus t squared you to the minus t plus two. And then you ve that becomes minus t you to the minus t. And I mean, just make that minus clear minus t e to the minus t minus the integral of V d u. So this becomes minus t squared e to the minus, and that is the type of there. It's a T minus t squared e to the minus t and then minus two t e to the minus t. And then when you integrate E to the minus T, that's minus e to the minus TSR minus two. You to the minus t so and then plus a constant of integration. So that is my integration. Now I want to find out how far it travels in the first tee second. So I'm looking at what's the integral from zero to t of t square and e to the minus T d t so that is going to be equal to, um I could just factor out minus ive had the minus t and that leaves me with t squared plus to t plus two. And this just needs to get evaluated from zero to t. So if I substitute a t Ah, and of that equation, it's just gonna be as it is minus e to the minus t t squared plus two t plus two. You substitute zero into that equation and you're gonna get minus E to the zero, That's what at minus one. And then when you substitute a zero m for that polynomial, you're going to get a positive, too. Eso this all turns into two, um minus e to the minus t t squared plus two t plus two. So that is the distance and the distance given him meters. So this is the meters that this particle travel in the first T seconds. So that's how far this particle travel to mine is. E to the minus t attempts t square plus to t plus two meters

All right. So a simple time dilation question here. Ah, you have to find a rest frame a lifetime. Now this particles which moves with a certain speed, close the speed of light and time dilation equation though we're using here is from Equation twenty six, Section one. And the book which states that ah, time elapsed in a rest frame is equal to the time elapsed in a moving fame. Times one minus Reece squared over C squared. And so Delta T and moving fame. We know that time is four point seven six times ten to the negative six seconds and then you most by that by one minus. V squared over C squared. So vee is two point seven times ten to the eighth C is three times tend to the aid. He's a boat in meters per second. You square the whole thing. And so the ten to the A and the meeting's for second car's cancel and what we get here is point gate one. And so we have this times four point Chris five nine. And so your answer is two point zero seven times times ten to the negative six seconds, which is roughly two microseconds

It's exercise. We're gonna talk about time dilation, so consider that we have to reference frames in the first one. We have to events that happened in different points in time, but at the same point in space to this to the difference of time between these two events, we give the name the proper time and we represented by this towel letter. And now consider that we have a second reference spring according to which, Okay, the first reference frame is moving with a certain speed Z notice that now the two events don't happen in the same place anymore. They happen separately. And the time t that is measured between these two events according to special relativity is equal to Tao. Time's gonna where gamma is one over the square root of one liners we squared, oversee square. Okay. And in our problem, we have a particle that's created in a lab that has a speed off 0.9 times the speed of locked and in the labs reference frame, the ah lifetime of the particle is equal to 2.3 pickle seconds. Okay, one pickle second is 10 to the minus 12. Sex and our goal is to find what is the lifetime of the particle in its own reference. For so we want to know the proper lifetime of the particles since the creation and the annihilation of the particle happened in the same place in the particles reference from Okay, so we have the tea is Tower times gamma. We want to find Tao, so I'm gonna isolated. Have the towel. I'm sorry. Tall is t divided by gamma. So this is tee times on over one minus v squared overseas. Where? Yeah, uh, so tha always 2.3 people seconds times the square root of one minus vv 0.9 c. So the oversee is your 0.9. So it's 2.3 times the square root of one minus 0.9 square. So towel is equal to one pickle second. So this is the proper time. Proper lifetime. The part of


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