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what does momentum mean the definition of momentum is the mass times the velocity so the formula is simple it's just M times V and why do we care about momentum we care about momentum because if there's no net external force on a system the momentum of that system will be conserved in other words the total initial momentum of that system would equal the total final momentum of that system so momentum will be conserved if there's no net external force and momentum is a vector that means it has components the total momentum will point in the direction of the total velocity and the momentum in each direction can be conserved independently in other words if there's no net force in the Y direction then the momentum in the Y direction will be conserved and if there's no net force in the X direction the momentum in the X direction will be conserved since the momentum is M times V the units are kilograms times meters per second so it's an example problem involving momentum look like let's say two blocks of mass 3 m and M head toward each other sliding over a frictionless surface with speeds 2 v + 5 V respectively and after the collision they stick together which direction will the two masses head after the collision to figure this out we can just ask what direction is the total momentum of the system initially since momentum is going to be conserved that'll have to be the direction of the momentum finally so the momentum of the three m/s is going to be the mass which is three M times the velocity which is 2 V so we get a momentum of 6 MV and the momentum of the mass M is going to be the mass M times the velocity which is negative 5 V momentum is a vector so you can't forget the negative signs which gives a momentum of negative 5 mV so the total initial momentum of this system would be 6 MV plus negative 5 MV which is 1 MV and that's positive which means the total momentum initially is to the right that means after the collision the total momentum will also have to be to the right and the only way that could be the case if these two masses join together is for the total combined mass to also move to the right what does impulse mean the impulse is the amount of force exerted on an object or system multiplied by the time during which that force was acting so in equation form that means that J the impulse is equal to the force multiplied by how long that force was acting and the impulse is going to be equal to the net force times the time during which that net force was acting and this is also going to be equal to the change in momentum of that system or object in other words if a mass had some initial momentum and ends with some final momentum the change in momentum of that mass P final minus P initial is going to equal the net impulse and that net impulse is going to equal the net force on that object multiplied by the time during which that force was acting and since impulse is a change in momentum and momentum is a vector that means impulse is also a vector so it can be positive and negative and the units are the same as momentum which is kilograms times meters per second or since it's also force times time you could write the units as newtons times seconds so what's an example problem involving impulse look like let's say a bouncing ball of mass M is initially moving to the right with a speed 2v and it recoils off a wall with a speed V and we want to know what's the magnitude of the impulse on the ball from the wall so the impulse J is going to be equal to the change in momentum the change in momentum is P final minus P initial so the final momentum is going to be the mass times the final velocity but this velocity is heading leftward so you can't forget the negative sign minus the initial momentum which would be M times 2v which gives a net impulse of negative three MV and this makes sense the net impulse has to point in the same direction as the net force this wall exerted a force to the left that means the impulse also points left and has a magnitude of three mV if you get a force versus time graph the first thing you should think about is that the area under that graph is going to equal the impulse on the object so if you graph the force on some object as a function of time the area under that curve is equal to the impulse just be careful since area above this time axis is going to count as positive impulse an area underneath the time axis would count as negative impulse since those forces would be negative why do we care that the area is equal to the impulse well if we can find the area that would equal the impulse and if that's the net impulse on an object that would also equal the change in momentum of that object which means we could figure out the change in velocity of an object so what's an example problem involving impulses the area under a graph look like let's say a toy rocket of mass 2 kilograms was initially heading to the right with a speed of 10 m/s and a force in the horizontal Direction is exerted on the rocket as shown in this graph and we want to know what's the velocity of the rocket at the time T equals 10 seconds to figure that out we'll figure out the area under the curve this triangle would count as positive area this triangle would count as negative area and since this triangle is just as positive as this triangle as negative these areas cancel completely and the only area we'd have to worry about is the area between 8 seconds and 10 seconds this is going to end up being a negative area since the height of the rectangle is negative 30 and the width of the rectangle is going to be 2 seconds this gives an impulse of negative 60 Newton seconds so if the impulse on this object is negative 60 Newton seconds that's going to equal the change in momentum of that object how much momentum did this objects start with the initial momentum of this object is going to be 2 kilograms times the initial velocity which was 10 meters per second to the right which is positive 20 kilogram meters per second so if the initial momentum of the rocket is positive 20 and there was a change in momentum of negative 60 the final momentum just has to be negative 40 or in other words since the change in momentum would have to be the final momentum minus the initial momentum which was positive 20 we could find the final momentum by adding 20 to both sides which would give us negative 60 plus 20 which is negative 40 what's the difference between an elastic and an inelastic collision what we mean by an elastic collision is that the total kinetic energy of that system is conserved during the collision in other words if a sphere and a cube collide for that collision to be elastic the total kinetic energy of the sphere plus the kinetic energy of the cube before the collision would have to equal the kinetic energy of the sphere plus the kinetic energy of the cube after the collision if the total kinetic energy before the collision is equal to the total kinetic energy after the collision then that collision is elastic it's not enough for the system to just bounce off of each other if two objects bounce the total kinetic energy might not be conserved only when the total kinetic energy is conserved can you say the collision is elastic for an inelastic collision the kinetic energy is not conserved during the collision in other words the total initial kinetic energy of the sphere plus cube would not equal the total final kinetic energy of the sphere plus where does this kinetic energy go typically in an inelastic collision some of that kinetic energy is transformed in the thermal energy during the collision while masses could bounce during an inelastic collision if they stick together the collision is typically called a perfectly inelastic collision since in this collision you'll transform the most kinetic energy into thermal energy and when two objects stick together it's a surefire sign that that collision is definitely inelastic so what's an example problem that involves elastic and inelastic collisions look like let's say two blocks of mass 2m and M head toward each other with speeds for V and 6v respectively after they collide the 2m mass is at rest and the mass M has a velocity of 2 V to the right and we want to know was this collision elastic or inelastic now you might want to say that since these objects bounced off of each other the collision has to be elastic but that's not true if the collision is elastic then the objects must bounce but just because the objects bounce does not mean the collision is elastic in other words bouncing is a necessary condition for the collision to be elastic but it isn't sufficient if you really want to know whether a collision was elastic you have to determine whether the total kinetic energy was conserved or not and we could figure that out for this collision without even calculating anything since the speed of the 2m mass decreased the kinetic energy of the 2m mass went down and since the speed of the M mass also decreased after the collision the kinetic energy of the mass M went down as well so if the kinetic energy of both masses go down then the final kinetic energy after the collision has to be less than the initial kinetic energy which means kinetic energy was not conserved in this collision had to be inelastic one final note even though kinetic energy wasn't conserved during this process the momentum was conserved the momentum will be conserved for both elastic and inelastic collisions it's just kinetic energy that's not conserved for an inelastic collision how do you deal with collisions in two dimensions well the momentum will be conserved for each direction in which there's no net impulse if there's no net impulse in both directions then the momentum in both directions will be conserved independently in other words if there's no net force in the X direction the total X momentum has to be constant there's no net force in the y-direction the total momentum in the Y Direction has to be constant so in other words if two spheres collide in a glancing collision the total momentum in the X direction initially should equal the total momentum in the X direction finally if there's no net impulse in that X direction and the total momentum in the Y direction initially of which there is none in this case would have to equal the total momentum in the Y direction finally if there's no net impulse in the Y direction so what's an example involving collisions in two dimensions look like let's say a metal sphere of mass M is traveling horizontally with five meters per second when it collides with an identical sphere of mass M that was at rest after the collision the original sphere has velocity components of four meters per second and three meters per second in the x and y directions we want to know what are the velocity components of the other sphere right after the collision so assuming there were no net forces in the X or Y direction in this case then the momentum will be conserved for each direction and since the mass of each sphere is the same we can simply look at the velocity components so if we started with five units of momentum in the X direction we have to end with five units of momentum in the X direction so the X component of the second sphere has to be one meter per second and since we started with no momentum in the vertical direction initially we have to end with no momentum vertically so if the first sphere has three units of momentum vertically after the collision then the second sphere has to have three units of momentum vertically downward after the collision which gives us an answer of D what's the center of mass mean the center of mass of an object or a system is the point where that object or system would balance and the center of mass is also the point where you can treat the entire force of gravity as acting the way you can solve for the center of mass is by using this formula you multiply each mass by how far that mass is from the reference point if there's no reference point specified you get to choose the arbitrary reference point which would designate where x equals zero you continue adding each mass times its position for positions to the left of the reference point those would count as negative positions and when you're done accounting for every mass in your system you divide by the total mass which would be all the masses added up and the number you get would be the position of the center of mass the center of mass is going to have units of meters since it's a location the location where the system or object would balance and the location where you can treat the entire force of gravity is acting something else that's extremely important to remember is that the center of mass of a system will not accelerate unless there's an external force on that system in other words the center of mass of a system follows Newton's first law if the center of mass of a system is at rest then even if the masses in that system exert forces on each other and move around the center of mass of that system will stay put until there's a net external force on the system and if the center of mass was initially moving to the right at some speed that center of mass will continue moving to the right of that speed even if the masses are moving in different directions until there's a net force on that system so it's an example problem involving center of mass look like let's say a remote control car of mass M is sitting at rest on a wooden plank also of mass M in the position seen here there is friction between the wheels of the car and the plank but there's no friction between the plank and the ice upon which the plank is sitting now the remote control car is turned on and off what would be a possible final position of the car and the plank now because the car is at rest and the plank is at rest that means the center of mass of this system is also at rest and since there's no net force on this system the center of mass is going to have to remain at rest where is the center of mass will the cars mass is at three the plank center is at five so the center of mass between the car and the plank would be at the location of four so to find the correct solution we just need to figure out which one of these also has the center of mass at for option a has the car at three and the center of the plank at three that put the center of mass at three meters but that can't be right the center of mass can't move there were no external forces on our system and the center of mass started at rest so it's got a remain at rest for option B the center of the car is at four the center of the plank is at three this would put the center of mass somewhere between three and four but again that can't be right we need our center of mass to be at the location for option C has the car at six and the center of the plank at four this would put the center of mass of the system at five that can't be right we need our center of mass at for option D has the car at five and the center of the plank at three that puts the center of mass at location 4 just like it was before so D is a possible solution
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