A white ball traveling at 2.0 m/s hits an equal-mass red ball at rest. The white ball is deflected by 25°and slowed to 1.5 m/s.
b. What percentage of the initial mechanical energy is lost in the collision?

Question

A white ball traveling at 2.0 m/s hits an equal-mass red ball at rest. The white ball is deflected by 25°and slowed to 1.5 m/s.
b. What percentage of the initial mechanical energy is lost in the collision?

Accepted Answer

Hey, everyone in this problem, we have a 3.5 kg sphere moving at 3.2 m per second, that collides with a second sphere of equal mass initially at rest. OK. So let's stop there for right now and just draw out what we have. So we have this initial situation. OK. We have two spheres and we're gonna draw one in blue one in red. Now the first sphere we're gonna call it sphere one, it has a mass which we'll call M one of 3. kg and it's moving at 3.2 m per second. So we're gonna draw it moving to the right towards the red sphere. We'll take right and up to be our positive directions. And so this is gonna have a positive velocity and that is equal to 3.2 m per second. OK. So V 0.1, the initial speed of sphere, one is 3.2 m per second. Now, we have our second sphere which we've drawn in red, that sphere has the same mass. So M two is also gonna be equal to 3. kg and it's initially at rest. So V not two. OK. The initial speed of sphere two is going to be zero m per second. OK. So that's all the information we have. Now, let's keep reading, we're told that the direction of the fi first sphere changes by degrees and its new speed is 2.5 m per second after the collision. OK. So drawing this, we're gonna draw this final scenario after the collision, we still have our two spheres. So we have our blue sphere which has the same mass as initially M one equals 3.5 kg. We have our red sphere again, the same mass M two equals 3.5 kg. And we're told that the direction of the first sphere changes by 20 degrees. Now we're gonna say that it changes upwards and I'm gonna come back and tell you why we can assume that or why you could assume it downwards either way is OK. OK. So from the horizontal, this makes a 20 degree angle in that speed, the F one is 2.5 m per second. And we aren't told anything about the final speed or velocity of this second sphere. Yeah. All right. So we've drawn our diagram. Now, the question is asking us to express the lost mechanical energy as a percentage of the system's mechanical energy before the collision. And the problem is giving us four answer choices, option, a 24.8 degrees or sorry, 24.8%. Option B 31.9%. Option C 52.1% or option D 40.6%. Now we're asked to calculate lost mechanical energy. OK. Remember when we're looking at energy, particularly kinetic energy, we use the speed and not the velocity. All we care about is the magnitude of that speed. We don't care about the direction. And so choosing this final speed of sphere, one that we drew in blue to be pointing upwards 20 degrees instead of t downwards degrees is not gonna change the final answer of the lost mechanical energy. OK? You may get different signs as you work through the problem. I intermediate steps. But that final answer is gonna work out just the same. Now, we're asked to calculate the lost mechanical energy. So what we wanna do is calculate the mechanical energy before the collision, the mechanical energy after the collision and compare the two, I recall that our mechanical energy is equal to our kinetic energy. K. What's the potential energy? You now, in this case, we have no gravitational potential energy. These spheres are not at a height and we have no spring potential energy either. We have no springs acting on this system. And so you was going to go to zero, that potential energy is zero. And we only have the kinetic energy to worry about and recall it, the kinetic energy is given by one half M V. Squared and, and you can see that B squared value, that's why the sign of our speed doesn't matter. OK? Because we're squaring it, we're gonna get a positive value no matter what. Yeah, if we want to calculate mechanical energy before an actor, we're gonna need the speeds and we don't know the speed of sphere two after the collision. So let's start there. Let's calculate that speed. Then we can calculate our mechanical energies. Now, we can use conservation of momentum here, we have no net external forces acting on our system. So we can use a conservation of momentum before and after the collision. Now we have motion in both the X and Y direction. So we're gonna have to consider conservation of momentum for both of those directions. We have the initial momentum in the X direction. P not X is equal to the final momentum in the X direction P FX. OK. Our momentum is made up of two things, two components because we have two objects in our system. So we have the initial momentum of sphere one in the X direction plus the initial momentum of sphere two in the X direction is equal to the final momentum of sphere, one in the X direction plus the final momentum of sphere two in the X direction. Now recall that momentum is equal to mass multiplied by velocity. So for each of these terms, we get the corresponding mass multiplied by the corresponding velocity M one V, not one X plus M two V, not two X is equal to M one V F one X plus M two V F two X. Now we're gonna substitute in our values. We know that sphere two is not moving initially, so its initial velocity is zero. And so the second term on the left hand side, M two V 02 X goes to zero. We don't need to worry about that. Now, we also know that these masses M one and M two are equal. OK. So every single one of these terms has that 3.5 kg in it. So we can divide by that mass the entire equation, then these masses M one and M two will divide it. So on the left hand side, we just have V not one X which is 3. meters for a second. And on the right hand side, we get V F one X, OK. Now the final velocity of sphere one in the X direction that's gonna be related to the angle through the cosine because we're looking at the adjacent side. And so we get cosign of 20 degrees multiplied by 2.5 m per second, that speed plus that final velocity of sphere two in the X direction. OK. So solving for that final speed in the X direction that we're looking for. The only unknown in this equation, we need to subtract cosine of 20 degrees multiplied by 2.5 m per second. OK? We can simplify that and we get that this final speed is 0. m per second. All right, I'm gonna put a box around this so that we don't lose it. We can come back to it when we need it. So remember we want to find the mechanical energy to do that. We need this speed. OK? We found the X component of the speed. Now we need to do the same in the Y direction. So looking at our conservation of momentum in the Y direction, we have the initial momentum in the Y direction. It is equal to the final momentum in the Y direction. OK. This is gonna follow the exact same pattern as with the X component. We have two objects in our system. So we have two things that contribute to each of these momentum P, not one Y plus P, not two Y is equal to P F one Y plus P F two Y. And you'll notice that on all of these momentum, I've included a vector symbol, momentum is a vector. The direction does matter when we're dealing with momentum, momentum mass multiplied by velocity. So again, for all of these terms, M one V, not one Y plus M two V, not two Y plus M one V F one Y plus M two V F two Y. Now we know that sphere two is not moving initially, same thing as in the X component. And so that second term on the left hand side goes to zero. But when we think about the Y direction sphere, one also has an initial velocity of zero. OK. It's moving completely in the X direction, it's moving to the right, so it has no Y velocity. So the entire left hand side actually goes to zero. And I have written a plus sign here instead of an equal sign. So I apologize. So we have an equal sign between those initial and final momentum. Now we have zero on the left hand side of our equation, on the right hand side, we can divide out those masses just like we did in the X component because they're the same, we divide by 3.5 kg throughout the entire equation. And on the right hand side, we're left with V F one Y plus V F two. OK. So this Y component that we're looking for V F two Y is going to be equal to negative V F one Y, it's gonna have the same magnitude as a Y component of sphere one but in the opposite direction. Now V F one Y OK. The Y component of sphere one after the collision is gonna be related to the angle through the sign because it's on the opposite side. And so this is equal to negative sign of degrees multiplied by 2.5 m. Per second, moving down to give ourselves some more space to work. We get that V F two Y is equal to if we work this out negative 0.855, 05 m per second. OK. And that's our Y component. So we have our X component, we have our Y component. Now we can calculate the speed V F two that we need to find our mechanical energy. So we have a positive X component. So this is moving to the right, we also have a negative Y component. So it's moving down, OK. Positive X component moving to the right negative Y component moving down. And if we connect these two, that hypo is going to be V F two, the speed we're looking for. Yeah, again, we're looking for the speed we want the magnitude, we don't care about the direction. And so we're gonna just use absolute values on all of these values. OK. So V F two squared, it's going to be equal to V F two X squared plus V F two Y squared. When we square them, we're gonna get the positive value. Anyways, we get V F two squared is equal to 0. m per second squared plus negative 0. m per second squared. OK. If we work this out on the right hand side, we take the square root, we're gonna get the positive and negative root. But again, what we care about is the magnitude. OK. So we're just gonna take the magnitude of that route that we get. We get that the magnitude of our speed V F two is equal to 1. m per second. All right. So we have this final speed of sphere two that we need for our mechanical energy calculations. OK? So now let's get to this mechanical energy calculation. And let's start with the initial mechanical energy. You need the mechanical energy we're gonna call eating mac. Not remember that we said there is no potential energy. So mechanical energy is just equal to the kinetic energy. So the initial mechanical energy is equal to the initial kinetic energy. It's gonna be the kinetic energy initially of sphere one, what's the kinetic energy initially of sphere two? Now sphere two is at rest initially it is not moving. So it has no kinetic energy. That second term goes to zero. And we're left with just k not one, the initial kinetic energy of sphere one recall kinetic energy is one half M V squared. So we have one half M one V, not one squared, we can substitute in our values here. One half multiplied by 3.5 kg, multiplied by 3.2 m per second, all squared which gives us 17.92 jewels. OK. So we have this initial mechanic mechanical energy of 17.92 Jews. Now let's look at the final mechanical energy so that we can look at how much was lost. OK. So we get final mechanical energy E Mac F Well, this is gonna be equal to the final kinetic energy since we have no potential energy which is equal to the final kinetic energy of sphere. One plus the final kinetic energy of sphere two, this is equal to one half M one V F one squirt plus one half M two V F two squared. Substituting in our values one half multiplied by 3.5 kg multiplied by 2.5 m per second squared. And then I'm just gonna add the second half of this equation underneath it. So we don't have to squish it in and we don't run out of room. We get one half multiplied by 3.5 kg multiplied by that speed. We found 1.2062 m per second all squared. And when we work this out, we get a final mechanical energy of 13.4836 Jools. OK. So we can see that we started with 17.92 Jews. We ended with 13.4836 jewels. So there was a loss of mechanical energy. Now, we want to find the percent of lost mechanical energy and express it um as a percentage of the systems of mechanical energy before the collision. OK. So what we're gonna do lost energy percentage is we're gonna take the difference of these two and we're gonna take the initial mechanical energy minus the final mechanical energy. We're gonna divide it by the initial mechanical energy because we're told that we want to take it as a percent of the energy before the collision. Ok. So we're gonna divide by that mechanical energy before the collision. I'm going to multiply all of this by 100 to get that percentage. Ok? So substituting in our values, we get 17.92 jewels minus 13.4836 Jules two divided by 17.92 Jews. We're gonna take all of that and multiply it by 100 and this is going to give us 24.76% right? And so the last mechanical energy as a percentage of the system's mechanical energy before the collision is 24.76%. If we compare this with their answer choices, we see that this corresponds with answer choice. A ok. If we round to three significant digits, we get 24.8%. Thanks everyone for watching. I hope this video helped see you in the next one.

A white ball traveling at 2.0 m/s hits an equal-mass red ball at rest. The white ball is deflected by 25°and slowed to 1.5 m/s. b. What percentage of the initial mechanical energy is lost in the collision?

Verified Solution

Key Concepts

Conservation of Momentum

Kinetic Energy

Energy Loss in Collisions