momentum
A quantity expressing the motion of a body/system that equals the product of mass and velocity
The inertia of motion;
A measure of how hard it is to get something that is moving to stop or change directions;
a measure of the force associated with motion
What is the relationship between force and momentum?
[1] the contact (collision) force equals the change in momentum divided by contact time.
[2] net force is the measure of how the momentum changes in a certain amount of time.
impulse
the change in momentum of an object;
a measure of the average force of contact applied for how much time(or duration) of contact;
impulse is the product of the average force of contact and contact time
What three things are important to remember about force associated with an impulse?
The force of contact is not constant. For example, when a bat hits a baseball the force starts out small, rises to maximum, and then drops off as ball leaves the bat.
Since the force is not constant, When we discuss the force associated with impulse we refer to the average force of contact.
the magnitude of the impulse delivered by a force during a time interval is equal to the area under the force vs time
What is the impulse-momentum theorem?
the impulse of force acting on an object equals the change in momentum of that object.
REMINDER: Frame of reference
It's very important to keep track of your velocities' directional sign of + and -.. it's safe to describe motion to the right(Eastward) as positive and left(westward) as negative.
How do boxing gloves or rolling with the punches lessen brain injury?
Brain injury is caused when it expediences a large and sudden force. Padded gloves extend the time delta t over which the force is applied to the head. For a given implulse (Favg)(time), if the time duration increases the average Force decreases. Gloved or non-gloved the force the boxer throws his fist doesn't change and neither will the speed, however the time can be altered.
What can reduce or eliminate the chance of dying in a car crash?
Collision time. According to Newton's third law the force of the crash is the same in each vehicle. However a seatbelt will extend the time over which the passenger experiences the collision thus lowering the force experienced.
isolated system
a a system where the only forces present are those between the objects of that system.
this means that no external forces are acting on the objects in the system.
law of conservation of momentum
For an isolated system, momentum is conserved(remains constant).
when no net external force acts on a system, the total momentum of the system remains constant in time.
equation for law of conservation of momentum
Proof of law of conservation of momentum
How is law of conservation of momentum derived from Newton's 3rd Law?
how is the law of conservation of momentum commonly misinterpreted? explain with a cheerleader jumping.
One may incorrectly identify the isolated system; A cheerleader jumping upwards from rest may appear to violate conservation of momentum because initially her momentum is zero and suddenly she is leaving the ground with a velocity. The flaw in this reasoning is that the cheerleader isn't an isolated system. In jumping, she exerts a downward force on the earth, changing it's momentum. This change in momentum isn't noticeable because the earth's mass is so large and it's velocity isn't appreciable visible. This is especially seen in recoil.
recoil
the backward momentum caused by the action and reaction force when objects initially at rest exchange momentum.
What type of problems require solvation via momentum?
How do you know not to solve via Newton's 2nd law or Work-Energy?
Problems involving collision, and Newton's third law.
One will know not to use Newton's second law because we don't know how the force will vary over time. Or information will not be given on the Force applied to the object or the acceleration the object expereinces
(a)During a collision what happens to momentum?(b) what is this called? (c)what happens to the kinetic energy?
(a)During any collision momentum is conserved. This means that the total momentum before the collision equals the total momentum afterwards as long as the system is isolated.
(b) impulse approximation
(c) The total kinetic energy generally is not conserved because some kinetic energy is converted into sound, internal energy, and work needed to permanently deform an object.
elastic collision
a collision where both momentum and kinetic energy are conserved.
inelastic collision
a collision where momentum is conserved but kinetic energy is not conserved.
perfectly inelastic collision
a collision where momentum is conserved, kinetic energy is not and the two objects stick together so their final velocities are the same.
equation for perfectly inelastic collision
ballistic pendulum
it is a device used to measure the speed of a fast moving projectile such as a bullet. The bullet is fired into a large block of wood suspended from some light wires. the bullet is stopped by the block and the entire system swings up to a height, h. perfect example of an perfectly inelastic collision.
kinetic energy equations for elastic head on collisions
Generally, when is it not ok to use the simplified kinetic energy equation to solve elastic collision problems?
In 2 or 3 dimensional collision problems
What happens when object of equal mass collide elastically?
The objects exchange velocities
glancing collision
a collision between objects in a plane (2 dimensional space)
What methodical plan should one follow to solve a glancing elastic collision problem?
(1) diagram (2)conservation of momentum (3) conservation of energy (4) solve simultaneously
(a)why is it necessary to consider and objects center of mass?
(b) define center of mass
(a)this point in our discussion of motion we have only consider the translational motion of an object. However when an object moves it can do so in many different forms (i.e rotational, vibrational, etc.) especially rotational. However, extensive observations of bodies in motion have shown that even if a body rotates( or several bodies move relative to one another) there is one point that moves in the same path that a singular article would if subjected to the same net force(or I think we could say if the particle only experienced translational motion and no rotational motion).
(b) this point is considered the Center of Mass.
Consider the motion of the figure. Diver A experiences only translational motion while Diver experiences translational and rotational motion. The Center of Mass experiences the same parabolic path, while other points in the rotating diver's path follow mo
How is the Center of Mass calculated? How is this different for objects that are two dimensional?
We can consider any extended body as being made up of many tiny particles. But we first consider this body to be made of two particles of mass m1 and m2. We choose a coordinate system so that both particles lie on the same straight line(doesn't have to be) at x1 and x2 respectively on the x-axis. If the two masses are equal, by calculation the CM will be the midpoint. If one mass is greater than the other, the CM will be closer to the larger mass.
If the objects are two dimensional then a separate calculation will need to be done in vertical (y direction) for each point. same equation will be used
How is the Center of Mass calculated? (Figure)
center of gravity
Is this concept different from center of mass? If so how
the point in the body at which the force of gravity can be considered to act.
we usually assume that the entire weight of the body acts at one point, the CG.
It is different from the center of mass but for all practical purposes we can consider the center of gravity and mass to be the same point.
a difference between these two points would happen if the body was so large that the acceleration due to gravity is different at different parts of the body
When calculating the CM for the human body, what is the significance of knowing the hinge points.
When a body part is bent this will happen at a hinge point. The Center of Mass will change as a result and so the calculation for the center of mass becomes a two dimensional problem.
Can the center of mass lie outside the body? give examples to support your point
Yes. a donut has a center of mass at the hole where no mass is found. a bent appendage on the body. a high jumper has a CM below the bar even though their body clears the bar.
thrust
reaction Force exerted on a rocket by the ejected exhaust gases which causes the rocket to accelerate.
Driving force behind a car vs a rocket
when a car moves, the driving force of motion is friction. The driving force is exerted by the road on the car as a reaction to the force exerted by the wheels against the road. however a rocket moving in space has no road or tracks to push against , so how does it move forward? Consider an open and closed chamber. If an explosion happens inside the closed chamber, the hot gas expands in all directions against the inner walls of the chamber. Because the sum of forces on the chamber is zero, the chamber doesn't move. Now suppose a hole is drilled in the bottom of the chamber, when the explosion occurs, the gasses will press against the chamber in all directions, but can't press against anything at the hole, where it simply escapes into space. Adding the forces in the spherical chamber now results in a net force upwards., opposite that in which the gas escapes. The wall on the rockets chamber exerts a force on the gas expanding against it. The reaction force of the gas on the wall then pushes the rocket upward.
Driving force behind a car vs a rocket (figure)
Explanation of rockets using momentum (pt1)
Explanation of rockets using momentum (pt2)
Why must a golfer or baseball player follow through on a swing?
To increase the time ensuring that the ball leaves with the maximum amount of velocity the bat and ball are in contact(force applied) ensuring that the ball leaves with the maximum amount of velocity and that the ball leaves the club(its source of force) at the furthest distance in the range of the swinger. Yes the force is reduced because of the increased length of time the club and ball are in contact but that's ok because the ball leaves the club further away and with a higher velocity than at the tee. Also when a golfer follows through he doesn't slow the club speed at the tee.
Why are boxing gloves, padded dashboards or airbags used?
To lengthen the time of contact thus lowering the force for a given impulse(velocity and mass). airbag or no airbag the velocity and mass of the object will not be different, but if you lengthen the time for that mass and velocity the force experienced is lowered.
Why does lengthening time in contact lower the force of a collision?
Let force equal the equation above. As t increases for a given change in velocity, F decreases. the shorter the t the higher F and concordantly the higher the acceleration.
what mistake did I initially make with net force = 0 for conservation of momentum to be true?
It's not the net force on the object it is net force on the system.
REMINDERS for Momentum and collision problems
Make sure we haven't confused system for an individual object within that system. Make sure we define what direction is considered positive and negative
QQ6.2 A boy standing at one end of a floating raft that is stationary relative to the shore walks to the opposite end of the raft, away from the shore. As a consequence, the raft (a) remains stationary, (b) moves away from the shore, or (c) moves toward t
(c) moves toward the shore
QQ6.3 A car and a large truck traveling at the same speed collide head on and stick together. Which vehicle experiences the larger change in the magnitude of it's momentum? (a) the car (b) the truck (c) the change is the same (d) impossible to determine
(c) the change in magnitude of the momentum is the same for both
QQ 6.4 An object of mass m moves to the right with a speed V. It collides head on with an object of mass 3m moving with speed V/3. in the opposite direction, If the two objects stick together what is the speed of the combined object, of mass 4m, after the
(a) 0
QQ 6.5 A skater is using very low friction rollerblades. A friend throws a Frisbee at her, on the straight line along which she is coasting. Describe each of the following events as an elastic, an inelastic, or perfectly inelastic collision between the sk
(a)perfectly inelastic collision
(b)inelastic collision
(c)inelastic. though the return velocity is equal in magnitude to what was initial velocity(which would indicate elastic), this velocity was not the result of a collision or bounce. It was the result of the person catching and throwing the Frisbee. When she catches and throws it, some kinetic energy deformation and generation happens respectively.
*QQ6.6 In a perfectly inelastic one dimensional collision between two objects, what condition alone is necessary so that all of the original kinetic energy of the system is gone after the collision? (a) The objects must have momenta with the same magnitud
(a) the objects must momenta with the same magnitude but opposite directions.
*CQ 1 A batter bunts a pitched baseball, blocking the ball without swinging. (a) Can the baseball deliver more kinetic energy to the bat and batter than the ball carries initially? (b) Can the baseball deliver more momentum to the bat and batter than the
(a) No. upon impact some kinetic energy is converted into sound, heat and friction. So the kinetic energy on the bat/batter after collision would be less than the original KE of the ball before. If collision is perfectly elastic the most the ball can deliver would be exactly what it initially had.
*(b) Yes. By bouncing from the object it strikes, it can deliver more momentum in a collision than it possess in its flight. It can deliver more momentum because when the ball bounces it can have some final velocity with a negative sign. therefore the range for maximum change in momentum(impulse) delivered is from -v to positive v, which would be twice the value of from 0 to v. For kinetic energy because the value of v is squared, if it left with some -v the value when squared becomes positive resulting in 0 KE transferred. the range of maximum change in KE delivered would be from v to a final v of 0. therefore it could deliver no more than what it initially had
CQ 2. America will never forget the terrorist attack on September 11, 2001. One commentator remarked that the force of the explosion at the Twin Towers of the World Trade Center was strong enough to blow glass and parts of the steel structure to small fra
Both the paper and the steel/glass structures experience the same Impulse(momentum) from the force of explosion. Due to the elasticity, paper experiences the force of explosion for a longer period of time which reduces the magnitude of force experienced. The steel/glass structure feels the full force because it has limited elasticity and cannot move with the force of explosion. Therefore the many remain intact.
CQ 3. In perfectly inelastic collisions between two objects, there are events in which all of the original kinetic energy is transformed to forms other than kinetic. Give an example of such an event?
If all the kinetic energy disappears, there must be no motion of either of the objects after collision. If neither is moving, the final momentum of the system is zero, and the initial momentum of the system must also have been zero. A situation in which this could be true would be the head on collision of two object having momenta of equal magnitude but opposite direction.
CQ4. If two objects collide and one is initially at rest, is it possible fro both to be at rest after the collision? Is it possible for only one to be at rest after the collision? Explain
No it's not possible for both objects to be at rest after collision. For momentum to be conserved, one object can be at rest but the other object must have some velocity for momentum to be conserved.
Yes it is possible for only one to be at rest afterwards, as previously discussed.
CQ 5, A ball of clay of mass M is thrown with a speed V against a brick wall. The clay sticks to the wall and stops. Is the principle of conservation of momentum violated in this example?
Initially, the clay has momentum directed toward the wall. When it collides and sticks to the wall, neither the clay nor the wall appears to have any momentum. Thus, it is tempting to (wrongfully) conclude that momentum is not conserved. However, the "lost" momentum is actually imparted to the wall and Earth, causing both to move. Because of Earth's enormous mass, its recoil speed is too small to detect.
CQ 6 A skater is standing on a frictionless ice rink. Her friend throws a Frisbee straight at her. In which of the following cases is the largest momentum transferred to the skater? (a) The skater catches the Frisbee and holds onto it. (b) The skater catc
The largest momentum transferred to the skater happens when the skater catches the Frisbee, holds it momentarily and throws it back to her friend. According to the LOCMomentum, momentum is conserved so the momentum of the Frisbee will be completely transferred to the skater. In scenario (c), the Frisbee goes starts with a velocity and after collision ends with a negative velocity. In scenario (b) the Frisbee starts with velocity and ends wih a velocity of zero. In scenario (a), the Frisbee starts with the same velocity but ends with a velocity significantly less due to the combined mass of the perfectly inelastic collision.. Scenario (c) has the greatest range of change in velocity from -v to v and will thus have the greatest Implulse or change in momentum. scenario (b) is second because the range of change in velocity goes from v to 0. this impulse is exactly half of scenario c. Lastly is scenario (a). The range of transfer goes from v to so smaller v but greater than zero. The difference( or range) would be smaller than that in scenario (b)
CQ 7. You are standing perfectly still and then you take a step forward. Before the step your momentum was zero, but afterwards you have some momentum. Is the conservation of momentum violated?
Before the step the momentum was zero, so afterward the net momentum must also be zero. Obviously, you have some momentum, so something must have momentum in the opposite direction. That something is Earth, the enormous mass of which ensures that its recoil speed will be too small to detect, but if you want to make Earth move, it is as simple as taking a step.
*CQ 12 Consider a perfectly ineleastic collision between a car and a large truck. Which vehicle loses more kinetic energy as a result of the collision?
the car losses more kinetic energy. The loss of energy is exactly equal to the final kinetic energy of the truck and since the trucks mass is larger when know it will be greater in value...Solve for the change in kinetic energy for each and you will discover this to be so. Remember that since the collision is perfectly inelastic the final KE of the car and truck individually will be the mass of both car and truck.
CQ 13 Your physical education teacher throws you a tennis ball at a certain velocity, and you catch it. You are now given the following choice: The teacher can throw you a medicine ball (which is much more massive than the tennis ball) with the same veloc
It will be easiest to catch the medicine ball when its speed (and kinetic energy) is lowest. The first option--throwing the medicine ball with the same velocity-- will be the most difficult because the speed will not be reduced at all. The second option, throwing the medicine ball with the same momentum, will reduce the velocity by the ratio of the masses. The third option, throwing the medicine ball with same kinetic energy, will also reduce the velocity, but only by the square root of the ratio of the masses.
CQ 15 In golf, novice players are often advised to be sure to "follow through" with their swing. Why does this make the ball travel a longer distance? If a shot is take near the green, very little follow through is required. why?
The follow-through keeps the club in contact with the lass as long as possible, maximizing the impulse. Thus, the ball accrues a large change in momentum than without the follow-through, and it leaves the club with a higher velocity and travels farther. With a short shot to the green the primary factor is control, not distance. Hence there is little or no follow-through, allowing the golfer to have a better feel for how hard he or she is striking the ball.
CQ 18 A large bedsheet is held vertically by two students. a third student, who happens to be the star pitcher on the baseball team, throws a raw egg at the sheet. Explain why the egg doesn't break when it hits the sheet, regardless of its initial speed.
The eggs velocity is going from a maximum value to zero just like when it hits a wall. However due to the elasticity(flexibility) or give in the sheet it takes a longer time to get to zero so the Force applied to the egg is considerably smaller than the force the egg would experience when it crashes into a wall.
REMINDER Solving for the equations for Kinetic Energy
When solving for initial and final kinetic energies make sure to solve for the entire system not just one or two objects within.
If more than two objects are colliding at a time do the have to collide at the same time for accurate calculations? why or why not
NO. Since momentum is conserved (not loss or gained for a system) after each individual collision momentum will be adjusted to reflect the new scenario and momentum is conserved each time regardless of when the collision happens.
P38 Four railroad cars, each of mass 2.50 E4 kg are coupled together and coasting along horizontal tracks at speed V1 toward the south. A very strong but foolish movie actor riding in the second car uncouples the front car and gives it a big push, increas
IF the actor did no work, the carts would continue to move along at the initial speed. However when he did work by pushing the cart he not only affected the speed of the individual cart, he effected the speed of the other three still coupled carts(the system) So if Work done equals the change in Kinetic Energy, the change in kinetic energy of the entire system must be considered.
A small sports car collides head-on with a massive truck. The greater impact force(in magnitude) acts on (a) the car, (b) the truck (c) neither the force is the same on both.
Which vehicle undergoes the greater magnitude acceleration? (d) the car, (e) the
(c) According to Newton's third law, the force is the same on both.
(d) The car because it's smaller mass. When we divide the Force/ mass to solve for acceleration, the acceleration value will be larger.