Lecture Notes (Momentum & Impulse)
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1 Lecture Notes (Momentum & Impulse) Intro: - earlier in the year, we introduced Galileo's Principle of Inertia and we talked how Newton used this idea to formulate his first law of motion - Newton discussed inertia in terms of objects both at rest and in motion; today we will discuss the concept of inertia in motion only - when you are dealing with both momentum and motion you are talking about momentum - momentum is inertia in motion; specifically, it is the product of mass and velocity Momentum = mass velocity p = mv - the unit of momentum is kg m/s - it is intuitive to say that an aircraft carrier is harder to stop than a plastic toy boat moving at the same speed; the carrier has more momentum than the toy boat - by looking at the definition of momentum you can see that a moving object can have a large momentum either by having a large mass or a large velocity or both - the aircraft carrier has more momentum traveling at the same speed as the toy boat by virtue of its larger mass - the toy boat could have as much or even more momentum as the aircraft carrier, but its velocity would have to be extremely fast
2 Impulse: - a change in momentum can occur if there is a change in mass, a change in velocity, or both - if the momentum of an object changes while the mass remains unchanged, which is usually the case, then it is due to a change in velocity - changes in velocity are due to accelerations; accelerations are due to net forces acting on objects - the greater the net force on an object, the greater the acceleration, the greater the velocity change, the greater the change in momentum - there is another important factor in changing momentum; time - apply a force to a crate on the ground for a brief amount of time and you get a small change in momentum; if you apply that same force for a longer amount of time, then you get a larger change in momentum - a long sustained force will produce a greater change in momentum than the same force applied briefly - in terms of momentum, both the amount of force and the amount of time during which the force act are important; the product of these two terms is called the impulse Impulse = force time interval Impulse = FΔt - whenever you exert a force on an object you also exert an impulse
3 - the resulting acceleration depends on the force; the resulting change in momentum depends on the force and the time in which the force acts Impulse-Momentum Equation: - the relationship of impulse and momentum is derived from Newton's 2nd law of motion; F = ma; then time interval of impulse is hidden in the acceleration variable (Δv/Δt) - if you rearrange Newton's 2nd law you get: Force time interval = change in mass velocity (momentum) FΔt = mδv Increasing Momentum: - if you want to increase the momentum of something as much as possible, you not only apply the greatest force possible, you also apply that force for as long as possible - for example, long range cannons have long barrels; the longer the barrel the greater the velocity of the shell - this occurs because the longer barrel allows more time for the force of the exploding gunpowder to act on the shell (increasing Δt) - the force that is exerted on most objects varies over time; for example, in the cannon the gunpowder applies a greater force initially and the begins to lessen as the shell goes down the barrel - so, in order to simplify things, we will use the average force acting on an object
4 Decreasing Momentum (Over a Long Time): - one intuitive way to discuss decreasing momentum is to imagine you are in a car out of control and you can either slam into a concrete wall or a haystack - it's obvious which solution is the better choice; the more important reason in physics is why this is the better choice - in either case, hitting the wall or the haystack, your momentum will be decreased by the same impulse (FΔt); remember impulse is the product of force and time, but does not mean the same force or same time - you have a choice; if you hit the haystack your time will be extended (the time during which your momentum is brought to zero) - this longer time corresponds to a lesser force; for example, if you extend the time by 100 times, then the force will decrease by if you hit the wall, the time will be reduced, so the force will increase
5 - people often use the knowledge of impulse to their advantage; for example, a wrestler thrown to the floor will relax their muscles in order to extend the crash into a series of smaller impacts - the wrestler will have his/her foot, knee, hip, ribs, and shoulder hit the mat in turn; this will increase the time of impact which reduces the amount of force on the body - a boxer will move his face backwards in order to increase the time of impact of an opponent's blow - ballet dancers prefer to dance on a wooden floor rather than a concrete floor, because the wooden floor has "give" Decreasing Momentum (Over a Short Time): - in certain situations it is advantageous to decrease your momentum quickly - when you decrease the time interval, the size of the force will increase; for example, a karate master will endeavor to hit a stack of bricks with a quick blow; this will maximize the force of the blow (See picture below)
6 Bouncing: - impulses become even greater if bouncing occurs - bouncing will amplify the impulse because it takes an additional impulse to throw an object back after collision - for example, take the case of a flower pot hitting you head; if the pot breaks upon colliding with your head it does some damage; if the pot bounces off your head without breaking, it would do even more damage to your body - your head has to provide an additional impulse to send the pot back off your head - the effect was known and used for mining in California during the gold rush; a man named Pelton created a water wheel with curved blades to maximize the impulse of the water on the wheel
7 Conservation of Momentum: - in order to accelerate an object you must apply a force on it; in order to change the momentum of an object you must apply an impulse on it - the forces that are applied to an object, in order to accelerate it, must be external forces, not internal ones - external forces originate from outside the object you are studying; internal forces occur within the object itself - for example, the molecular forces within a rock do not change the momentum of the rock; similarly, if you are sitting in a car and push on the dashboard, you cannot change the momentum of the car; these are internal forces - if you fire a bullet from a gun, the forces present are all internal forces so the momentum of the system (gun and bullet) will not change - using Newton's third law (action/reaction), we know that the force exerted on the bullet will be equal in size and opposite in direction to the force of the bullet on the gun
8 - likewise, the time these forces are acting is equal; this results in equal momentum but in opposite directions - the recoiling rifle has just as much momentum as the bullet - both the gun and the bullet have gained momentum, but since they are in opposite directions, they cancel out leaving the total momentum of the system at zero - this is exactly the same momentum the system started out as; no momentum was gained or lost - this example shows us two important qualities of momentum: 1. momentum is a vector quantity 2. momentum is conserved (only in a closed, isolated system) Elastic Collisions: - momentum is conserved in collisions; that is the total momentum of the system of colliding objects is unchanged before, during, and after the collision - if you have one billiard ball hitting another ball at rest, head on, then the original moving ball will come to rest and the second ball which was at rest will move with the speed of the colliding ball
9 - this type of collision is called an elastic collision; the objects collide without deformation or the generation of heat Inelastic Collisions: - in inelastic collisions, the colliding objects are severely deformed (they may even stick together) and they generate heat (See the two pictures below)
10
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