Chapter 7

elastic collision

A collision in which colliding objects rebound without lasting deformation or the generation of heat. when the kinetic energy before and after the collision are found to be the same collision of two billiard balls is the closes example. BUT, even when kinetic energy is not conserved (unelastic), total energy is ALWAYS conserved.

inelastic collision

A collision in which the colliding objects become distorted, generate heat, and possibly stick together. no conservation of kinetic energy kinetic energy that is lost is changed into other forms of energy, like thermal energy

Momentum in everyday

A fast moving car has more momentum than a slow moving car of the same mass A heavy truck has more momentum than a small care moving with the same speed. The more momentum an object has, the harder it is to stop it and the greater effect it will have on another object if it is brought to rest by striking it. A football player is more likely to be stunned if tackled by a heavy linebacker running at top speed than by a small wide receiver running at the same speed or a heavy linebacker moving at a slower speed.

Force and Momentum

A force is required to change the momentum of an object, whether to increase it, decrease it, or change its direction. For a heavy linebacker moving a top speed: 1. We can increase the momentum of the linebacker by pushing him faster 2. We can decrease the momentum of the linebacker by creating more friction on the ground he runs on 3. we can change the linebacker's direction by having another linebacker physically push him to the side.

isolated system

A system that can exchange neither energy nor matter with its surroundings The sum of all internal forces within the isolated system will be zero because of Newton's third law If there are external forces, forces exerted by objects outside of the system, then the total momentum of the system will not be conserved. .

Impulse (physics definition)

Amount of change in momentum- application of force over a period of time Impulses tell us that we can get the same change in momentum with a large force acting for a short time, or a small force acting for a longer time. This is why you should bend your knees when you land; why airbags work; and why landing on a pillow hurts less than landing on concrete. *Impulse = F*t* The concept of impulses is useful when dealing with forces that act during a short time interval, such as when a bat hits a baseball

Collision According to Newton's Laws

During a collision of two objects, there is a force exerted by object A on object B at any instant is F. By Newton's third law, he force exerted by object B on object A is represented as -F during collision time on other external forces are acting

Conserved Momentum

If no net external force acts on a system, the total momentum of the system is a conserved quantity. In the real world external forces do act: friction on billiard balls, gravity on a tennis ball, ect. Therefore, we want our observation time to be small. Momentum is conserved as long as we measure the momentum of each ball just before the collision and immediately after the collision.

No Conservation Example

If we take our system as a falling rock only, it does not conserve momentum because an external force (gravity) from the earth *accelerates* the rock and changes its momentum. However, if we include earth in our system, the total momentum of the rock plus earth is conserved.

Law of Conservation of Momentum

The rule that in the absence of outside forces the total momentum of objects that interact does not change. *The total momentum of an isolated system of objects remains constant* That is, the vector sum of all individual momenta before collision will be equal to the vector sum of all individual momenta after the collision. ` If the net force on the system is zero, the total momentum doesn't change

Law of Conservation of Momentum

The rule that in the absence of outside forces the total momentum of objects that interact does not change. Useful when dealing with a system of two or more objects that interact with one another: like collisions of ordinary objects or nuclear particles

solving ballistic pendulum

There are two parts to solving these kinds of problems: part 1 we look at the collision of the bullet with the wooden box (pendulum). Since we are looking at a collision, momentum is conserved and we can use an equation for the conservation of momentum. *mv = v' (m+M)* - before the collision (left) measures only the bullet. After the collision (right) measures the wooden box and bullet. Part 2: we look at the swinging of the pendulum after its been hit with the bullet. the movement of the pendulum is subject to a net external force (gravity tending to pull it back to its vertical position when it swings upward). Therefore, we cannot use the conservation of momentum equation for this part. BUT, we can use conservation of mechanical energy because gravity is a conservative force (KE + PE) = (KE + PE) *if there had been motion of the pendulum while the bullet was colliding with it, there would have been an external force working (gravity) during the collision, so the conservation of momentum would no have been valid in this case*

inelastic collisions in one dimension

Two objects collide inelastically (kinetic energy not conserved) A completely inelastic collision is one where the objects stick together afterwards, so there is only one final velocity for all objects Although momentum is the only quantity conserved, we can still write an equation to solve for the unknown final velocity v'

collisions in two dimensions

conservation of energy and momentum can also be used to analyze collisions in two or three dimensions. one example of this situation is when a moving object (projectile) strikes an object at rest (target). Like in a game of pool, projectiles from radioactive decay, or high energy accelerators knowing the masses and initial and final velocities is not enough; you need to know angles as well in order to find the final velocities.

inelastic collisions

kinetic energy not conserved with inelastic collisions some of the kinetic energy is lost to thermal or potential energy. It may also be gained during explosions, as there is the addition of chemical or nuclear potential energy usually macroscopic (large particle) collisions are inelastic. completely inelastic: two objects stick together (putty) even though the kinetic energy is not conserved, total energy is always conserved and the total vector momentum is always conserved

Conservation of Momentum (rocket)

momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system, and remember to include the mass of the rocket. The mass of the fuel includes the mass of the high speed gases leaving the exhaust nozzle

Conservation of Energy and Momentum Collisions

momentum is conserved in ALL collisions In elastic collisions, when two objects are very hard and no heat or other energy is produced in the collision, then the total kinetic energy is the same after the collision as before. For the brief moment during which the two objects are in contact, some of the energy is stored momentarily in form of *elastic potential energy*

Location of CM

on a coordinate line, the CM will be closer to the particle with the larger mass. If two particles have the same mass, the CM will lie directly in between them

center of mass

real objects undergo rotational and other types of motion, not just translational. A diver undergoes only tranlational motion when they dive straight body in A diver undergoes rotational and translational motion when they do a summer sault. motion that is no translational is reffered to a *general motion*

general motion

the general motion of an object can be considered as the sum of the translational motoin of the center of mass, plus rotational, vibrational, or other forms of motion about the center of mass. Sum off all different CM points

A bowling ball and ping pong ball are moving toward you with the same momentum. If you exert the same force to stop each one, for which is the stopping distance greater?

the ping pong ball work energy theorem KE = W = FD. Thus, stopping distance increases with KE. The ball with less mass has the greater KE

center of gravity

the point around which an object's weight is evenly distributed. the center of gravity is the point where gravitational force can be considered to act. It is the same as the center of mass as long as the gravitational force does on vary among different parts of the object. the CG can be found experimentally by suspending an object form different points. The CM need not be within the actual object. For example, a doughnuts CM is the center of the hold.

center of mass

the point in an object that moves as if all the object's mass were concentrated at that point if we draw the arrows for the divers motion - translational only, the arrows only point up - rotational and translational, the arrows change direction there is one point on the arrow that moves in the same path a single particle would, this point is the center of mass the same parabolic path the summer sault diver follows would be the same direction that the projectile particle follows

linear momentum

the product of the mass and velocity of an object Momentum is represented by the symbol "p". If we let m represent mass of an object and v represent its velocity we get our equation

A small beanbag and a bouncy rubber ball are dropped from the same height above the floor. they both have the same mass. Which one will impart the greater impulse to the floor when it hits?

the rubber ball the impulse delivered by the ball is twice that of the beanbag the rubber ball doesn't stop when it hits the ground it continues to bounce. This means that the final velocity for the ball is not zero like it is for the beanbag.

center of mass and translational motion

the total momentum of a system of particles is equal the the product of the total mass of the particles and the velocity of the CM. the sum of all the forces acting on a system is equal the the total mass of the system multiplied by the acceleration of the system's center of mass.

Elastic Collisions in one dimension

two objects colliding elastically. We know the masses and initial speeds Since both momentum and kinetic energy are conserved, we can write two equations using mA, vA, mB, vB; and the prime version (after collision) of each. this allows us to solve for the two unknown final speeds. v'A & v'B for any elastic head on collision, the relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision no matter what the masses are.

elastic collisions in two dimensions

if the collision is elastic, we have three independent equations we can solve for three unknowns These variables can include the final velocities of both objects along with one of their angles. - one angle must be given already since we can only solve for three unknowns, not 4.

Solving problems - momentum conservation and collisions

1. choose the system 2. consider whether a net external force acts on the system. if it does, be sure the time interval is short enough 3. draw a diagram of initial (before collision) and final (after collision) 4. coordinate system. for head on collision, use only x axis. 5. apply momentum conservation equation. one equation for each component (x,y,z) 6. if collision is elastic, you can also write down a conservation of KE equation 7. solve for unknowns

Total Momentum of a System of Particles

For a system of particles, the total momentum is the vector sum of the momenta of each individual particle

Conserved Quantities

In addition to the law of conservation of energy, other quantities are found to be conserved Linear momentum angular momentum electric charge

Vectors

Momentum and Velocity are both vectors The direction of velocity and momentum is the same & the magnitude of momentum is represented by the equation. A reference frame must be specified since BOTH velocity and momentum depend on it.

A bowling ball and a ping pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

Same time for both We know that Force = momentum/time, so Time = momentum/force Because F and P are the same for both balls, it will take the same amount of time to stop them

An open cart rolls along a frictionless track while its raining. As it rolls, what happens to the speed of the cart as the rain collects in it? (Rain falls vertically)

The cart slows down Since the rain falls vertically, it adds no momentum to the box, thus the box's momentum is conserved. However, since the mass of the cart slowly increases with the added rain, its velocity has to decrease

Conservation of Momentum Example

The head on collision of two billiard balls: Assume the net external force on this system of the two balls is zero. The only significant forces during the collision are the forces that each ball exerts on the other Although the momentum of each ball changes as a result of the collision, the SUM of their momenta is found to be the same as before the collision. the total vector momentum of the system of two colliding balls is conserved: it stays constant.

Newton's Second Law of Motion

The rate of change of momentum of an object is equal to the net force applied to it. we can readily derive the familiar form of the second law (F = MA) from the equation of net force from momentum. This is a more general statement for Newton's second law, than the previous one because it includes a situation in which the mass may change. A change in mass may occur in certain circumstances, such as for rockets which may lose some of their mass as they expel burnt fuel.

Collisions and Impulse

When a collision occurs, the interaction between objects involved is usually far stronger than any external forces. We can ignore the effects of the other forces during the brief time interval of the collision. During a collision of two ordinary objects, both objects are deformed because of the large forces involved. When the collision occurs, the force each objects exerts on the other usually jumps from zero at the moment of contact to a very large force within a very short time, and then rapidly returns to zero again.

Ballistic pendulum

a devise used to measure the velocity of a fast-moving projectile such as a rifle bullet a device used to measure the speed of a projectile, such as a bullet. The projectile (bullet) of a certain mass is fired into a large block of wood or material of a mass that is suspended in the air. As a result of the collision, the pendulum (wooden block) and projectile (bullet) swing up to a maximum height "h".