And x = 0 when t = determine the velocity; position, and total distance particle is defined by the relation J = _ 3 mlsec Ifv = 6 mlsec The acceleration of a given ...

The motion of a particle is defined by the relation $x=6 t^{4}-2 t^{3}-12 t^{2}+$ $3 t+3,$ where $x$ and $t$ are expressed in meters and seconds, respectively. Determine the time, the position, and the velocity when $a=0 .$

We're told that the motion of a particle is fine by the relationship. Exit goes 60 to the fourth minus two cubed minus 12 T squared plus three t plus three. What accent here expressed in meters and seconds. Determine the time, the position and velocity when the acceleration is zero. That's the first weekend. Take derivatives to get our velocity and celebration from our position. And then we can figure out the time when, um, our acceleration is zero and we get that that's happened that when t equals 2/3 of a second and we can plug that back in to our position and our velocity and we get the position is 0.26 meters and the velocity is minus 8.56 meters per second and obviously the acceleration at that 0.0

So in this problem were given a equation of motion. And we want to find the time, the position and the velocity when the acceleration is equal to zero. So we have Yeah, this formula for the equation of motion. Yeah. And we're going to go ahead and take the time derivative So d d t of the position, she'll give us the velocity. Yeah, yeah, yeah. So that would be 24 t cute. Minus 60 squared. I'm just 2040 plus three. And then if we take TV of duty of the velocity will give us the acceleration. 72 t squared and it's 12 t minus 24. So we want to know when the acceleration is zero so we can set that equal to zero and factor things out to simplify it. So that would give us 12 times 60 Square Dynasty Managed to is the zero. Yeah, and we see that this is a quadratic equations. We can go ahead and solve that, which gives us two solutions. So to use to third seconds. And thi is one half seconds, but we can go ahead and reject that one. We're going to stick with this one. Yeah. 0.667 seconds. And here you just want to go with the one that's the most physical. So at T equals two third seconds. We're just gonna plug that into our original formula for X. So plugging that into our first equation of motion Well, give us a position of 0.259 m. And now we want to plug two into our equation for velocity. So we'll call this one two, three. So now that we have this by plugging it into equation one, we're going to put this into equation, too. We'll get that. The velocity at this time is negative. Key point 56 m per second. And from above, we know that the time was 0.667 seconds. When the acceleration is ego

Seen this problem were given a you relation that defines the emotion of a particle. And we want to determine the time, the position and the velocity when the acceleration is zero. So we're giving that X is equal to 60 to the fourth minus two t cubed minus 12 t squared plus 32 plus three mhm. So the first thing that we want to do is take the derivative with respect. Um, dx DT so we can find the lawsuit. So during that would give us mhm and then to find the acceleration we take the derivative of that should give us Yeah. So when a is equal to zero Yeah. So I need your t squared minus 12. T 24 is equal to zero, so and we can go ahead and factor this so it's easier to solve. So, b 12 times, sixties squared minus T minus two is equal to zero. Then we can, uh, see that there's two solutions using the quadratic formula. That would be t is two thirds No, and to use one half of a second. But we're going to go ahead and reject that one and go with this solution So at this time, we're just going to plug that in to our first equation. That's what we started with. So plugging this for T until equation one consulting give us a exposition of 0.259 leaders. Then we're also going to plug this into our equation for velocity. We'll call it equation, too. Okay, which will give us a velocity at that time negative 8.56 m per second. And all that negative sign tells us that they are going in the opposite direction as we've defined our access.

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