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The position of a ball as a function of time is given by$x=(5.0 \mathrm{m} / \mathrm{s}) t+\left(-10 \mathrm{m} / \mathrm{s}^{2}\right) t^{2} .$ (a) What are the initial position,initial velocity, and acceleration of the ball? (b) Plot $x$ versus $t$ for$t=0$ to $t=2.0 \mathrm{s}$ (c) Find the average velocity of the ball from$t=0$ to $t=1.0 \mathrm{s}$ (d) Find the average speed of the ball between$t=1.0 \mathrm{s}$ and $t=2.0 \mathrm{s} .$

(a) The Answer is $[\mathrm{zero}],[5 \mathrm{m} / \mathrm{s}]$ And $\left[-20 \mathrm{m} / \mathrm{s}^{2}\right]$(b) SEE SOLUTION(c) $-5 \mathrm{m} / \mathrm{s}$(d) 25 $\mathrm{m} / \mathrm{s}$

Physics 101 Mechanics

Chapter 2

One-Dimensional Kinematics

Motion Along a Straight Line

Cornell University

Rutgers, The State University of New Jersey

University of Washington

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Okay, So in this problem, we have a function that describes the position of a particle. The function in his ex if course five D minus Stan T Square. Okay, so the first item of the problem, we need to discover the initial velocity, the initial position and the acceleration of this particle. Since this is ah, second degree function, we know that this should be accelerated movement. And we know that the generic immigration that describes, uh, position of a particle is x zero bolos V zero t plus a divided by two T square. Okay, so let's look to the initial position. The initial position is when the time equals zero. So sporting T equals zero. We're going simply have initial position off zero meters. And what about the initial velocity? The initial velocity? We can come there. That's weird. In here we have X equals five key minus stand key square. So if we look here, we see that the zeroes needs to be five. So the initial velocity is just five meters for a second. What about the separation? Do the same thing. We just compare the genetic function we have to the position with the one we have in this particular problem. So Dan minus Stan needs to be close to a divided by two, which means that a is going to be minus trying t meters per second squared. This is the answer to the first item. The second item. We need to do a lot of dysfunction. Okay, let's put a function here. We have Let's see, ex grows five D minus Dan de square, and we need to blot the graph between the Times T CO zero and T Coast through. So thinks this is a second degree function. We know that the behavior of dysfunction of this graph should be a horrible and we just need to discover what is going to be the position and t because two seconds. Let's find this. We have to position it's gross five times, two miners 10 times for So this is just minus dirty minor starting in the position. Okay, so let's see. So we need to remake his graph because we're not gonna be able to a lot this graph in minus dirty with the X is like this. So let's see, That's pretty exes. Uh, here we know this is a second degree function and dysfunction has aken captivity negative because off the constant follows the teeth square. So we know that this function begins in zero. It grows up in two minus dirty. So we're going to have a lot like this. This is the position the plot in T equals two seconds. In the third item, we need to discover the average velocity between the time deco zero anti equals one. So let's see average. The last city is just the displacement divided by the difference in time difference. In time. We already have. It's just one second and the displacement? Well, let's see. We just need to discover what is the final position and t close one and the position Antico zero. We already know. So the position and t it goes one going to be five times one minus 10 times one. This is simply minus five. So the average velocity in this interval is just minus five minus. Mine is you know. So the answer to the third item is the average speed, the average velocity. I want to be minus five meters per second in the last item. We need to discover the average speed between worm into seconds. So let's do the same thing again. We already know the position. Yeah, T equals one. But what is the position and the equals? Two seconds. So the position you're going to be two. Sorry. Five times to minus 10 times four. So this is equal to 30 minus dirty meters. So the average velocity, it's going to be attacks divided by daughter T. This is simply minor. Sturdy, minus minus five. Just going to come positive, divided by one. So this is just minus 25 meters per seconds. That's different. Arrested to the problem. Thanks for watching.

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