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Find the magnitude of the acceleration vector for the position function:r(t) = (166,- ~25t7 _ 3t.40*) (Distance meters, time seconds)Find = {Peed at t = ZseC Ora mo...

Question

Find the magnitude of the acceleration vector for the position function:r(t) = (166,- ~25t7 _ 3t.40*) (Distance meters, time seconds)Find = {Peed at t = ZseC Ora moving object with position vector r() = (t? 102-^ Zvt + 2. Vy' ["(4) (26 - | 2e3 Speed 41+2vi?)= < ? , 2 , 2 | Iveju = [z?, 2",@Find tbe velociry fuociiod t) girea tat alt) =< 0 5857,1(0) =<6-2,4>. JaU)a <el _ ,,7(ost)- ? v6)-<4,2,47=7<4,2,47 +3 19,-1)+2 2 =4,-2 ,5)70-(ke"+ $ , $-2,-al+57

Find the magnitude of the acceleration vector for the position function: r(t) = (166,- ~25t7 _ 3t.40*) (Distance meters, time seconds) Find = {Peed at t = ZseC Ora moving object with position vector r() = (t? 102-^ Zvt + 2. Vy' ["(4) (26 - | 2e3 Speed 41+2 vi?)= < ? , 2 , 2 | Iveju = [z?, 2",@ Find tbe velociry fuociiod t) girea tat alt) =< 0 5857,1(0) =<6-2,4>. JaU)a <el _ ,,7(ost)- ? v6)-<4,2,47=7<4,2,47 +3 19,-1)+2 2 =4,-2 ,5) 70-(ke"+ $ , $-2,-al+57



Answers

The position of an object as a function of time is given by $\vec{r}=\left(3.2 t+1.8 t^{2}\right) \hat{\imath}+\left(1.7 t-2.4 t^{2}\right) \hat{\jmath} \mathrm{m},$ with $t$ in seconds. Find the object's acceleration vector.

Key were given a position vector, We're going to find the velocity, speed and acceleration of the object. So first to find velocity, let's take our first derivative and we'll do that of each component. So if two T. In the I direction, one in the J. And then when we bring down that three halves that will multiply by two. So we're just going to have a three T to the one half power in the K direction for our speed, we're going to take each of those components and square them, add them together and take the square root. So we'll get that four t squared plus one plus 90. There's nothing to clean up there. So we'll just leave it like that. So now we can go ahead now we can go ahead and take the next derivative. It's two in the I direction DJ piece goes away and then we have a three halves t to the negative one half in the K direction. Just clean that up. You don't want to really keep your negatives. Um exponents in there, put it on the bottom and make it a positive exponent

They give us a position function are of teeth is equal to Let's see. The first component is 25. The second component. Let's see, That's a negative 16 T Square plus 15 team plus fire. And now they want us to find the velocity and acceleration functions well. The Velocity is just the first derivative of position and acceleration, which is the second derivative. Let's go ahead and compete. Derivative, however, actor valued function. Let's see, the director of the first component is just 25 of the second component. It's negative. 32 t last 15. Nice spiral And then let's see. Second derivative will get a constant zero. When you take that tribute and let's see, we'll get negative there, you to hear nothing.

That the acceleration let's see is equal to zero negative 32 and the initial velocity is given by the Vector 50 The initial position is given by the vector 0 16 500 16. That looks good. So how we're going to find the position where you need to take derivatives one step at a time? Because the velocity can be found by taking the amputee ra tive of the acceleration function and then finding the constants. So let's see. We'll end up with zero here. Well, it could be some constant that will account for that Yet on negative 32 t here and then we're adding an arbitrary constant director. See? So let's see, we know that only plug in zero, we get 50 Soloviev zero. Well, the first back there is zero doubt. So it becomes a zero vector and then we have plus are unknown. Vector C is equal to 50 So if we added that our velocity function actually is and these directors component ways five on negative 32 t plus that zero Now we confined our position function by taking the let's see and see derivative of our velocity function with respect to time. So we'll end of with, let's say, a component wise five t negative 16 t squared, and we're going to add an arbitrary constant again, What's 80. But we know that the initial position only black and zero. Well, actually, this part gets here right out. So we just end up with the zero factor, plus our unknown vector as equal to zero common 16. So if we wanted to find our position function, we can just add this specter and our constant rector component wise. We just sold for that constant rector and sequel to what they gave us. So then we have, Let's see, five Teen Plus Zero that's just five teen on the negative. 16 T squared, plus 16. Okay, that's their position function.

The acceleration function is equal to is T sign of teeth. The initial velocity is given by two negative six. The initial position is given by 10 4 Well, we confined our velocity by taking the anti derivative of acceleration, the respect of time. We'll get some unknown constant that we gets all for with our initial condition, wandered with t squared over two in this time and then a negative co sign of teeth in this term, plus an unknown constant record. See, let's see. Looks good. Now we can plug in zero. Well, what happened? Swimming plug in zero. We end up with zero on our first component here, and co sign of zero is just one. But there's negative in France, we end up with a negative one, and then, plus are unknown elector or give us to negative six. Well, that's all for see. We can add the specter to the other side and end up with two negative five. Once we had So our velocity function. We can actually just add this specter and what we solve for C component ways. Teoh. Let's see t square over to Plus Two and negative co sign of teeth mine right And we can take the anti derivative one more time when searching for our position function on TV for the position. The 18 we'll end of west. Let's see T Q over six last duty and on the side of a grand that with negative sign anything minus five t and then we can't forget about our new constant director that we need to add. And so now let's make use of our initial information on my blood and zero. But you this component becomes zero on the second component becomes zero to you this time. So we just have plus deep is equal. Teoh 10 and four military Well, I just says that are constant. Rector is 10 for So to find our position function, we can add these two vectors component ways are constant and what we integrated to get. So we'll end up with seeing cute over six plus two Team Woz 10. And in the second component, we're gonna end up with negative sign guilty minus five of deep plus four. So you get awesome


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