Fred (mass 60 kg) is running with the football at a speed of 6.0 m/s when he is met head-on by Brutus (mass 120 kg), who is moving at 4.0 m/s. Brutus grabs Fred in a tight grip, and they fall to the ground. Which way do they slide, and how far? The coefficient of kinetic friction between football uniforms and Astroturf is 0.30.

Video transcript

Hey everyone in this problem during a warm up in a Velodrome, we have Charles an inattentive cyclist moving at a velocity of 12 m per second. I hat collides with Albert who's moving negative 10 m per second. I hat Charles and Albert and their bikes become entangled and begin to slide together on the horizontal floor. The friction coefficient between the cyclists bike system in the Velodrome is 0.15. We're asked to find the direction of motion after the collision and the distance traveled on the floor until they stop. The combined mass of Charles in his bike is 72 kg. And for Albert and his bike, it's 70 kg. We're given four answer choices. Option A 0.2 m in the negative X direction. Option B 0.45 m in the positive direction. Option C 12 m in the negative X direction and option D 24 m in the positive X direction. Now let's start just by writing out the information we were given. So starting with the masses, we have the mass of Charles and his bike. We're gonna call MC just 72 kg and we have the mass of Albert and his bike ma which is 70 kg. We're gonna think about this problem in two kind of parts. So we have pre collision and before the collision, we have a speed for Charles VNA C of 12 m per second. And we're gonna take to the right to be positive. So that's our positive I hat direction. And so this is just gonna be a positive 12 m per second and we have the initial velocity of Albert which is going to be negative 10 m per second. Now, post collision, what happens? Well, we have a speed, we're gonna call that V AC the velocity of Albert and Charles. Now we don't know what that is, but we want to find that out because we're interested in the direction of motion. So we need to look at the sign of this velocity to figure out that direction. So that's the first thing we need to figure out and we have just one speed because Charles Albert and their bikes all become entangled and move together. OK? They slide together. And so that's just gonna be one speed. Now, when we think about the slide and we have this sliding face, we have an initial speed for this slide that's actually gonna be equal to V AC. OK? That speed immediately after the collision is gonna be the speed that they start their slide with. OK. So V knot, if we're thinking about this slide is gonna be that speed V of Albert and Charles together. Now the final speed is going to be zero m per second. We want to know how far they travel until they stop. Ok. So that final speed is gonna be zero m per second because they come to a stop. We don't know the acceleration, we wanna find the distance travel delta X. We don't know the amount of time it takes. Now let's start with this speed V AC. OK. We're gonna need that in our equation when we're working with the slide to find the distance traveled. And we're gonna need it for the direction of motion. So we can go ahead and use that. And what we wanna do is think about conservation of momentum and we know that we have a conservation of momentum. And so the initial momentum P is gonna be equal to the momentum, we'll call it PAC when Albert and um Charles come together right after that collision. Now our initial momentum is made up of two things. We have the momentum from Albert and we have the momentum from Charles. So this is gonna be the momentum initially of Charles pull us the momentum initially of Albert, which is equal to that momentum when they collide and come together. Now we call that momentum is mass multiplied by velocity. So for each of these, we get the corresponding mass multiplied by the corresponding velocity MC multiplied by V not C plus MA multiplied by V, not A is equal to that total mass of Charles and Albert MC plus MA multiplied by the speed that they're traveling immediately after the collision V AC. Now we can substitute in what we know. We actually have all the information we need to solve for V AC. That's the only unknown in this equation. So substituting in our values, we have 72 kg multiplied by 12 m per second plus 70 kg, multiplied by negative 10 m per second. And notice that I've included the negative here. When we're talking about momentum, it is a vector value. So the direction matters just like it does when we're using velocity. And so we need to include that negative to indicate the direction on the right hand side, we get 72 kg plus plus 70 kg. And all of that is multiplied by that velocity V AC that we're looking for simplifying on the left and the right on the left, we have 164 OK. Our unit here is kilogram meter per second. And on the right hand side, we have 142 kg multiplied by V AC. OK. To solve for V AC, we're gonna divide both sides by 142 kg. The unit of kilogram will divide out and we're gonna be left with meters per second, which is what we want for velocity. And we get 1. m per second. So we found the final velocity and this velocity is positive. OK. That indicates that we're going in that positive direction, the direction we've specified as positive, which is to the right or the positive I hat direction. OK. So we know that we're moving in the positive X direction. So if we look at our answer choices, we can already eliminate two options and we can eliminate option A and we can eliminate option C because they have the wrong direction. OK. So we've done step one, we've figured out the first thing that this question was asking. Now we need to figure out the distance traveled on the floor. If we look at the information we have about this slide, we now have the initial speed von we know V F, we want to find delta X. In order to find delta X, we need one more known value. We need to have three known values to use our kinematic equation to find delta X. Well, we were given information about the coefficient of friction here. So let's think about the forces we have and use that to find the acceleration. Now, if we think about the forces, we know by Newton's law that the sum of the forces and in this case, in the extraction is going to be equal to the mass multiplied by the acceleration. If we draw a free body diagram of this system, OK. We have motion going to the right. And so the friction is going to oppose that motion. Yes, we have our friction horse pointing to the left. We have the normal force pointing up and the force of gravity pointing downwards. So if we're looking in the X direction, the only force we have acting is this friction force, it's acting in the negative X direction. And so we can write that negative F K, it's equal to the mass multiplied by the acceleration F K. Can you recall the force of kinetic friction is equal to mu the coefficient of kinetic friction multiplied by N the normal force. So we have negative mu K N is equal to ma. Now, we need to look in the Y direction to find this normal force end. Now the sum of the forces in the wide direction. OK. This is gonna be equal to zero because we're in equilibrium. Hey, this Charles Albert, they're not moving up and down, they're just sliding across the floor in the next direction. So the, some of the forces in the Y direction is going to be zero in the positive Y direction. We have that normal force and in the negative Y direction, we have that force of gravity. So N minus F G is equal to zero, which tells us that the normal force is equal to the force of gravity. And recall that the force of gravity is given by mass times gravitational acceleration G. So we have N S equal to M G. We can use that in our equation for the X direction, we have negative mu K multiplied by M G is equal to ma dividing both sides by the mass. We get that the acceleration is equal to negative mu K multiplied by the acceleration due to gravity G. And we know the value of both of these. So we can substitute that in, we get that our acceleration is negative 0. multiplied by 9.8 m per second squared, which gives us an acceleration of negative 1. m per second squared. And that is the acceleration, we find that it's negative which makes sense because the bikers are gonna be slowing down and coming to a rest. So we have this negative acceleration that is gonna do that. OK. Now, if we look at our information for our slot, we now know V not V F and A that's three knowns, we can use those to find the unknown delta X. We're gonna look at our kinematic equations and choose the one that does not include time. T because we don't have information about the time. And that's not what we're looking for. That equation is gonna be V F squared is equal to V knott squared plus two A delta X. So we have zero is equal to 1.155 meters per second squared plus two multiplied by negative 1.7 m per second squared multiplied by delta X. Yeah, we're gonna move this delta X term to the left hand side as we try to isolate delta X. And we're gonna simplify. On the left hand side, we get 2.94 meters per second squared multiplied by delta X. Then on the right hand side squaring this initial speed, we get 1. m squared per second squared. And one more step dividing by 2.94 m per second squared. The unit we have meters squared per second squared, divided by meters per second squared. We're gonna be left with just a unit of meter, which is what we want for distance. So that's great. And we end up with 0. m. And that is the distance traveled by those bikers before they come to a stop. If we compare this to the answer choices we were given, we see that they are rounded to two significant digits. So if we round our answer to two significant digits, we had 0.45 m in the positive X direction which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.