A ball shot straight up with kinetic energy K₀ reaches height h. What height will it reach if the initial kinetic energy is doubled?

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Hey, everyone in this problem, a bullet is fired vertically with an initial kinetic energy of knot jewels. If it reaches a maximum height of eight m, what maximum height will it reach? If the bullet's initial kinetic energy is increased by a factor of four, consider VNA as the magnitude of the initial velocity of the bullet. We're given four answer choices. Option A V not squared divided by G option B four V knot squared divided by G option C two V knott squared divided by G In option D 1/2 v knot squared divided by G. Now we're gonna start, we're giving some information about this bullets kinetic energy as well as height. So let's recall, we have conservation of mechanical energy. The conservation of mechanical energy tells us that the initial kinetic energy plus the initial potential energy is going to be equal to the final kinetic energy plus the final potential energy. OK. So the initial mechanical energy equals the final mechanical energy. Now I've written the subscript not and then 01 OK? Or F and then a one and the one is going to indicate that first bullet, OK? The one that we know the maximum height and the initial speed and all of that information. Now what is kinetic energy? Well, kinetic energy recall is one half M the squared. Now, our potential energy, the potential energy here is going to be just the potential energy due to gravity. OK. The gravitational potential energy because the bullet is at a particular height and this is given by M G H, this is the same for the final case. So we have one half M VF- one squared plus um G each F one. Now, initially, the bullet is going to have no height. OK. We're gonna consider that the height that it's fired from is height zero. So the bullet has no gravitational potential energy. That term goes to zero. Now, when the bullet reaches a maximum height, It is going to momentarily stop at that maximum height before it starts to come back down. That means for a moment, it's going to have a speed of 0m per second at that maximum height. And so the final kinetic energy at that maximum height is going to be zero. OK. So now we're left with this one half M V knot squared, that's our initial kinetic energy. And we're actually told that that we're gonna call knot. So we have K knot on the left hand side and on the right hand side, we have M multiplied by G multiplied by H OK. We're told that it reaches a maximum height of H. So we have M multiplied by G multiplied by H. Now that's all the information we know about this initial situation and we can't simplify or do anything else. So let's move on to the second part of this problem and see what we can figure out here. We have the same thing, we have the conservation of mechanical energy. So we have the initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy. Now the initial kinetic energy of the second bullet is going to be four times that of the first bullet. OK. We're told that the kinetic energy initially is increased by a factor of four. So our knot two value is actually gonna be equal to four K knot. We then have the gravitational potential energy M G H 02. And again, this gravitational potential energy initially is going to be zero because the bullet starts at height zero, there's no initial height. This is equal to that final kinetic energy one half M VF two Squared. Once again, when the bullet reaches a maximum height, its speed will be zero. And so this final kinetic energy is zero. And then we have plus M G HF two. OK. That's spinal height two. And that's what we're trying to find out. This HF two value is the maximum height of this second bullet. This bullet where we've increased the initial kinetic energy. Bye. So let's just write what we have. OK, without all these markings. So we have four knot Is equal to m multiplied by g multiplied by HF two. Now we want to solve for HF two, we can go ahead and do that. We have the HF two Is going to be equal to four K divided by M G. Now, I wanna make a note here before we keep going. The answer. Solutions want us to write this in terms of V knot and G, OK. We could also write this in terms of H K K not divided by M G. If we go back to this first case, notice that that's just equal to H A H is equal to K knot divided by M G. So the final height of this second bullet or the maximum height of the second bullet is four times the maximum height of that first bullet. OK. So we've increased the initial kinetic energy four times and the maximum height has increased four times. OK. Now let's write this in terms of V knot and G and compare to our answer traces. Now we know that K knot is equal to one half M V knott squared. So this is equal to four multiplied by 1/2 M V not squared. And we divide by M G, the mass will divide out, we can simplify this coefficient four multiplied by a half, gives us two. And so we have two V not squared divided by G. And that is the answer we were looking for. If we compare this to our answer choices, we found that the maximum height of the bullet when the initial kinetic energy is increased by a factor of four will be two V knot squared divided by G which corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.

A ball shot straight up with kinetic energy K₀ reaches height h. What height will it reach if the initial kinetic energy is doubled?

Verified Solution

Key Concepts

Kinetic Energy

Gravitational Potential Energy

Conservation of Energy