A 50 g marble moving at 2.0 m/s strikes a 20 g marble at rest. What is the speed of each marble immediately after the collision?

Video transcript

Hey, everyone in this problem, we have two cars that collide with each other. Car A has a mass of 1000 kg and is moving towards car B with a velocity of 20 m per second. Car B has a mass of 800 kg and is at rest. We're asked, what is the velocity of each car immediately after they collide? OK. We're told to assume this is a one dimensional collision that obeys the conservation laws of momentum and mechanical energy were given four answer choices. ABC and D and each of them have a different velocity for A and for car B, we're gonna start by drawing a little diagram of what we have. So initially before the collision, we have car A and we're gonna just draw the car as a box. This is car A and we know that car A has a mass which we'll call ma of 1000 kg. Now, we're told it has a velocity of 20 m per second. We're gonna say it's moving to the right. So if it has a velocity of 20 m per second, that means that we're taking right to be our positive direction And so the velocity initially of A V not A is going to be equal to 20 m per second and it is moving towards Carby, I'm gonna draw Carby as a box too. Car B is at rest. So the initial velocity of car B V not B, it's going to be zero m per second. And we're told that car B has a mass which we'll call M B of 800 kg. All right. So this is the initial situation after they collide, we're trying to figure out what is the velocity of each car. So we don't have information about either of those things. We know that those blocks are gonna keep the same mass. So we're told that this collision obeys the conservation laws of momentum in mechanical energy. OK. So let's start with the conservation of momentum, conservation of momentum tells us that the initial momentum P knot is going to be equal to the final momentum P F. Well, we have two objects in our system that we need to consider that contribute to the momentum. We have car A and we have car B. OK. So our initial momentum is gonna be made up of the initial momentum of car A P. Not A, what's the initial momentum of car B PB? And that's gonna be equal to the final momentum of car A P FA plus the final momentum of car B P F B. Now recall momentum is mass multiplied by velocity. So for each of these terms, we've written, we're gonna break them down into the corresponding mass multiplied by the corresponding velocity. Yeah. So we have ma multiplied by V, not A plus M B multiplied by V, not B. On the left hand side. And on the right hand side, we get MA V fa plus M B V F B. And what we're looking for is those final velocities V FA and V F B. And those are the two values we're looking for everything else in this problem or in this equation, we have, we actually know. So let's go ahead and substitute in what we know. So the massive car a kg and its initial velocity is 20 m per second. Now, car B has an initial velocity of zero m per second. So this entire second term on the left-hand side is gonna go to zero. Then on the right hand side, we have the massive car again, 1000 kg multiplied by V fa plus the massive car B 800 kg multiplied by V F B. All right. So what we're gonna wanna do here is we're gonna wanna write one of these velocities in terms of the other. And we're gonna have a second equation that comes from our conservation of mechanical energy. So we're gonna wanna do substitution with the two equations and the two unknowns. OK. Those two velocities. So let's simplify, let's rate the speed or the velocity of car A in terms of car B OK. So on the left hand side, we have 20,000 kilogram meters per second. On the right hand side, we can't simplify any further. We have 1000 kg multiplied by V fa plus 800 kg multiplied by V F B. So isolating for V fa we're gonna move the 800 kg multiplied by V F B to the left hand side by subtracting. And then we need to divide by that 1000 kg. And we get that is equal to 20,000 kilogram meters per second minus 800 kg multiplied by V F B all divided by telegrams. And if we simplify this, we have V fa the velocity of car A after the collision is going to be equal to 20 m per second minus 0.8 multiplied by the final velocity of car B. And we're gonna call this equation one. OK. So this equation came from our conservation of momentum. Now we're gonna switch over to our conservation of mechanical energy. We're gonna get a second equation and then we're gonna substitute one into the other in order to solve for these two unknown values. So moving to our conservation of mechanical energy, OK. Same thing as moment we have that the initial mechanical energy which is the initial kinetic energy plus the initial potential energy is equal to the final kinetic energy plus the final potential energy Now, in this case, we have the cars on the ground, we have no gravitational potential energy acting, we don't have any springs present in our system. And so the potential energies are actually gonna be zero both initially and finally. And so what we really have here is a conservation of kinetic energy. OK. So we have the initial kinetic energy knot A, what's the initial kinetic energy of car B cannot not be, is equal to the final kinetic energy of car A K fa plus the final kinetic energy of car B K F B. Now recall that kinetic energy is one half mass multiplied by the velocity squared. OK. So we have one half multiplied by ma multiplied by V, not a square plus one half multiplied by M B multiplied by V not B squared. And this is equal to one half MA V fa squared plus one half M B V F B scored. All right. So we have this equation written now we're gonna start substituting in values and just like we did with conservation of momentum that initial velocity of car B is zero. And so the second term on the left hand side is going to go to zero. This leaves us with one half multiplied by the mass kg multiplied by the initial speed of car A 20 m per second squared. This is equal to one half multiplied by kg multiplied by V fa squared. And we're gonna substitute in equation one here in equation one, we had that V FA is equal to 20 m per second minus 0.8 V F B. So now we have 20 m per second minus 0.8 V F B all squared. OK. Plus one half multiplied by kg, multiplied by V F B squared. And what you'll notice now is that the only unknown in this equation is V F B OK. We used our conservation of momentum to write V fa the final velocity of car A in terms of car B. Now we've used our conservation of mechanical energy, we've substituted in that value and now we only have one which we can solve for. So on the left hand side, we simplify, we get 200,000 and our units are gonna be kilogram meter squared per second squared. On the right hand side, we get 500 kg, we can expand this square of our V fa OK. When we do that, we have 400 meter squared per second squared minus 32 m per second multiplied five V F B plus 0.64 V F B squared. And then we have that final term. And so we add 400 kg multiplied by V F B squared. Simplifying some more. On the right hand side, we're gonna leave the left-hand side alone for right now, 200,000 kg meter squared per second squared. Is equal to 200, kilogram meter squared per second squared. OK. We're expanding these brackets. So we're multiplying 500 kg into every term inside of these brackets minus 16,000 kilogram meters per second, multiplied by V F B plus kilograms multiplied by V F B squared. OK. And this final term takes into account the V F B squared we had inside of the brackets as well as this extra term that we added at the end. All right, we're getting close. We need to simplify a little bit more. We wanna move everything over to one side. So we move everything over to the right hand side. We have zero on the left hand side, we subtract 200,000 kg meters squared per second squared from both sides. And that is going to eliminate that 200,000 on the right hand side. And so what we're left with is 720 kg multiplied by V F B squared minus 16,000 kilogram meter per second multiplied by V F B. Now we want to solve for V F B. So let's go ahead and factor and we have some expression equal to zero. We factor this, we get that zero is equal to V F B multiplied by 720 kg multiplied by V F B minus 16, kg meters per second. So we get our first solution from this first term and that's just gonna be that V F B is equal to zero m per second. Ok? And we know that that was that initial speed. So what we're interested in is this other speed and the other speed is gonna be when the second term is equal to zero. So we have 720 kg multiplied by V F B minus 16,000 equal to zero. And it tells us that V F B is going to be 16, kg meters per second, divided by 720 kg, which gives us a final speed of 22. m per second. And that is gonna be the final speed of car beat. So we found the final speed of car B 22.22 m per second. Remember that we wanted final speed for car A and car B. So let's go back up to equation one where we had the speed of car A written in terms of the speed of car B. Now we can substitute in the value that we found for car B. So we have that the final velocity of car A is equal to 20 m per second minus 0.8 multiplied by that final velocity of car B which was 22.22 m per second. Ok? And if we simplify this, we get that the final velocity for car A is equal to 2.22 m per second. So we found after the collision, the final velocity of car A is 2.22 m per second and the final velocity of car B is 22.2 m per second, which corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.