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A ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d / 2 .$ Neglecting subsequent motion and...

Question

A ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d / 2 .$ Neglecting subsequent motion and air resistance, its speed $v$ varies with the height $h$ above the ground as :

A ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d / 2 .$ Neglecting subsequent motion and air resistance, its speed $v$ varies with the height $h$ above the ground as :



Answers

A ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d / 2 .$ Neglecting subsequent motion and air resistance, its speed $v$ varies with the height $h$ above the ground as :

So the bullet's going straight up. We'LL call ball one. Its position is going to be given by its velocity the initial on time. But it's one half prime for Nate. See where and the bullets dropped from a height Age ball two. This position is modelled by age. It is one half I'm for it. And you where? No, we want to find the time with the two balls collide. So we start there positioning auctions equal to each other. Why one is equal to white, too. And doing this. You see that the initial times time before throug h as the one half nine point eight square terms. Can't this gives us a time of It's survived the initial. Now we want to find the value of H in terms of being out and g such of the insulin. The balls collide the first balls of the highest point in motion. Okay, so the balls of the highest point of emotion that the apex of its trajectory and see their going up or down therefore philosophies zero. So if its velocity is zero, remember, this is the first ball that the ball that were throwing up you can use this equation. Find a velocity zero. The initial velocity is the same as in the previous part. He's known for Nate, but let's get T two. The initial org Most are actually with nine point eight, but we only want to write it in terms of B i n G. Where you want to find agent terms of the initial envy. So let's set these two equations time equal to each other, and we find that which you could be initial square bye bye great to be in it.

Christian number 73 The height after disagreement is Eun Bi di function s okay now our part eighties while fast is the more moving two seconds after there after being thrown. So we need to take dead already. So your velocity is equally true. Divided the off as an s. These negatives, Ext. Empty square Blessed 64. So, Daddy, with you off discoveries too deep and daddy ready Off the knees one. So here, aggression off the city started to D 64. Now take the music Will do. I do. But you're really easy called negative 32 into 64. So here we easy. Quite well, cereal that these after two seconds. The speed of the bullies Ciro Week for a second. Okay, No, But be that is how long after the body strong does it hurt? It's maximum height. So for the maximum hide take real music Will do. Siegel here 32 last 64 is Ecuador cereal 30 de is a full 64. So hear music would be go. So that is after TZ cordel through second Wait three days. ISS maximum height. No find that height. So your acidic a little negative. 16 to square last 64 into two. So gooseberries four and 64 into two is 1 20 Hey, so here s is Acquittal 64. Hey, that is hard. Maximum, Right. That is our maximum height. Dropped the ball At T's record. Two seconds. Thank you.

For this question are asked to consider two bolts. One ball is dropped straight downwards from a height H and then another. Balls thrown straight upwards from a initial height of zero with an initial velocity B not were asked to find a time in which they collide well, the height in which they collide. We can call that age subsea, and for the first ball that's gonna be equal to the height in which it was dropped. H Plus, there's no initial velocity. So, uh, this V not times the time T term is zero because there's no initial velocity. It was starting at rest, plus 1/2 times acceleration due to drop gravity g times the time T squared and for the second ball, the time in which they collide or the position which they collide. H subsea. It's the same as the first ball, but it's equation is vey not times the time t um, because it starts out with an initial velocity, but at H value of zero so that value isn't included, plus 1/2 times gravity times T square, so we can set these two equations equal to each other and salt for tea. So what we have is a tch. Plus 1/2 GT squared is equal to V not times the Time t plus 1/2 GT squared. Well, obviously the 1/2 GT squared cancel. You couldn't just subtract those out of both sides of the equation, so that's gone. And that's gone. We can solve for Time T when we find that tea is simply equal to the height in which the first ball was dropped, divided by the initial velocity of the other girl. When we come box set in as their solution for a part, B wants us to determine an expression for the height in terms of the initial velocity and gravity. So for Part B, we know that the final velocity of the ball here in this situation because we're considering when the first we're considering that this collision is gonna happen at the highest point of the motion of the first ball. So at the highest point of the motion of the first ball, the final velocity squared is equal to zero. But the equation for velocity is vey not squared. Um, plus two times gravity times the height H. So now putting H selling for H here we find that H is equal to V not squared over to G. So we did what we were asked to do. We found H in terms of V not in gravity, g. So we can box that in Is their solution to part B. That's the final part of our question.

Okay, so we know that in this problem we have a ball that was throw up works with initial speed to zero. Then, after reaching its maximum height, let's go. Height, age. The second ball was thrown upwards again with the same initials speech. So let's see. The problem wants to know in what position the balls will cross paths. So we know that this is a H divided by two. Let's call this the T one is the total time that we need to the boat make his movement. Okay, so this we have something like this. Okay, so what is the movement off the ball? One that put movement of ball one as a bloom off blue movement. The movement of blow off all one is going to be half of a parable that begins in position. Hey, JJ, then go stranger to the time one. It's not a very good credible, but it is a parable. And what is the movement off the ball? Two movement of both, who is also going to be a parable. But it's going to be this type off. Horrible. So as we can see the position when the boss cross paths is right here. So it's above age, divided by two through. The answer for this question is just I move Age divided by to transfer distressing. Just need to know I was going to be a movement in the graph, so


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