Momentum

I. Impulse-Momentum Theorem

Momentum 1. Defined Momentum is defined as “mass in motion” Momentum is a vector (there is a direction associated with it). Momentum is a measure of both mass and velocity. If the velocity of an object does not change, the momentum remains unchanged as well (Newton’s First Law of Motion)

click here for a introductory video on 2. Calculating The momentum (p) of a single object is the product of its mass and its velocity: p = mv There is no special unit for momentum. It's just the product of the units in the formula: kg-m/s click here for a introductory video on momentum from Bill Nye!

Momentum If the velocity of an object does not change, the momentum remains unchanged as well (Newton’s First Law of Motion) p = mv If momentum of an object changes, then it s is a result of a change in velocity ∆p = m∆v

1 Which has more momentum? A large truck moving at 30 m/s B small car moving at 30 m/s C Both have the same momentum. answer A Momentum is given by p=mv, both the truck and car have the same velocity so the difference in their momentum's are dependednt on the mass of the object and a truck has a greater mass then the car, therefore the truck has a greater momentum

2 What is the momentum of a 20kg object with a velocity of +5.0m/s? answer p = mv p= (20)(5) p= 100 kg-m/s

p = mv p= (20)(-5) p= -100 kg-m/s 3 What is the momentum of a 20kg object with a velocity of -5.0m/s? answer p = mv p= (20)(-5) p= -100 kg-m/s

p = mv v = p/m v= (-15)(5) v = -3m/s 4 What is the velocity of a 5.0kg object whose momentum is -15.0 kg-m/s? answer p = mv v = p/m v= (-15)(5) v = -3m/s

5 What is the mass of an object whose momentum is 35 kg-m/s when its velocity is 7.0m/s? answer p = mv m = p/v m= (35)(7) m= 5 kg

B. Impulse 1. Defined If we call the amount of momentum that we start with "pi" and the amount we end up with as "pf“, and if no momentum is added to or taken away from a system then pi = pf There is only one way to change the momentum of a system. If an outside force is used to change the velocity (accelerate) of the object, the object will have a change in momentum The outside force that changes momentum is called an Impulse (I) So if an outside force acts on a system, the general equation for the momentum of a system becomes: pi + I = pf

B. Impulse 2. Calculating Momentum is a conserved property. The momentum of an object remains unchanged unless and outside force acts on it. The amount the momentum changes is equal to the amount of impulse provided. What is impulse? Both the Conservation of Momentum and the concept of Impulse follow directly from Newton's Second Law: F = ma We can derive a mathematical expression for Impulse by examining acceleration….

C. Relation between Impulse and Momentum The formula for Impulse is I = F t The units for Impulse are either N-s or kg-m/s which are equivalent. And since I = p The Impulse delivered to an object, or system, is exactly equal to the change in its momentum.

8 An external force of 25N acts on a system for 10s. What is the magntude of the impulse delivered to the system? answer I = F t I = (25N) (10s) I = 250 N*s

9 In the previous problem, an external force of 25N acted on a system for 10s. We found that the impulse delivered was 250 N-s. In that case, what is the magnitude of the change in momentum of the system? answer I = p p = 250 N*s

B 10 The momentum change of an object is equal to the ______. force acting on it B impulse acting on it C velocity change of the object D object's mass times the force acting on it answer B

11 Air bags are use in cars because: A increase the force on the dashboard B increase the time of impact in a collision C decrease the momentum of a collision D decrease the impulse in a collision answer B By increasing the amount of time during the collision you reduce the force that you experience to slow you down, this helps to prevent injury, because the change in momentum remains the same, therefore by increasing the time you reduce the force.

impulse applied to the car 12 A car crashes into a concrete wall. The same type of car crashes into a padded wall at the same speed as the first crash. What is the difference between the 2 crashes? A change in momentum B impulse applied to the car C impact time D both B & C are true answer C During the collision you still experience the same change in momentum, meaning you also experience the same impulse, the only difference now is that the impact time has increased and the force is reduced.

13 In order to increase the final momentum of a golf ball, we could: A not change the speed of the golf club after the collision B increase the force acting on it C increase the time of contact between the club and ball D all of the above answer D By hitting it harder, applying a great force, and increasing the impact time would all result in an increase in the final momentum of the golf ball, that is if we increase the force and the contact time remained the same, and if we increased the contact time and the force remained the same

14 An external force acts on an object for 0.0020 s. During that time the object's momentum increases by 400 kg-m/s. What was the magnitude of the force? answer

* 15 A 50,000 N force acts for 0.030 s on a 2.5 kg object that was initially at rest. What is its final velocity? answer

Collisions and Impulse During a collision, objects are deformed due to the large forces involved. We can determine the relationship between the force, the time it acts and the change of momentum (often velocity) of the object by using our definition of impulse:

Law of Conservation of Momentum in Collisions These equations tell us that if no external force acts on a system, it's momentum will not change. For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2. We're going to look at three types of collisions. In all cases, momentum is conserved. In one case, Elastic Collisions, energy is conserved . In the other two cases, of Inelastic Collisions, energy is not conserved...some of the energy is transformed into heat, bonding, etc.

II. The Law of Conservation of Momentum During a collision—any collision--, measurements will show that the total momentum does not change: A B mAVA mBVB the prime means "after" A B mAvA + mBvB = mAvA' + mBvB' A B mAVA' mAVB' x

Conservation of Energy and Momentum in Collisions B VA VB Approaching A B Collision Momentum is conserved in all collisions. Collisions in which kinetic energy is conserved as well are called elastic collisions. This means the velocities are the same before and after the collision. Those in which it is not are called inelastic. A B VA' VB' elastic A B VA' VB' inelastic

C 16 In _______ collisions momentum is conserved. Elastic Inelastic B Inelastic C All answer C

A 17 In ______ collisions kinetic energy is conserved. Elastic B Inelastic C All answer A

Conservation of Momentum More formally, the law of conservation of momentum states: The total momentum of an isolated system of objects remains constant. VA = 4.5 m/s A B VB = 0 m/s Before collision After collision V' = ?

* A 13,500 kg railroad freight car travels on a level track at a speed of 4.5 m/s. It collides and couples with a 25,000 kg second car, initially at rest and with brakes released. No external force acts on the system. What is the speed of the two cars after colliding? Let's choose the first car's direction as positive. answer m1v1 + m2v2 = m1v1' + m2v2' v2 = 0 amd v1' = v2' = v' m1v1 = m1v' + m2v' m1v1 = (m1+ m2)v' v' = m1v1 / (m1+ m2) v' = (13500 kg)(4.5 m/s)/ (13500 kg+ 25000 kg) v' = (60750 kg-m/s)/ (38500 kg) v' = 1.6 m/s in the direction of the original car's motion

Inelastic Collisions M l vM = 0 m v With inelastic collisions, some of the initial 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 energy. A perfectly inelastic collision is one where the objects stick together afterwards, so there is only one final velocity. M+m l v' h

* A cannon ball with a mass of 100 kg flies in horizontal direction with a speed of 800 m/s and strikes a ship initially at rest. The mass of the ship is 15,000 kg. Find the speed of the ship after the ball becomes embedded in it. answer m1v1 + m2v2 = m1v1' + m2v2' v2 = 0 amd v1' = v2' = v' m1v1 = m1v' + m2v' m1v1 = (m1+ m2)v' v' = m1v1 / (m1+ m2) v' = (100 kg)(800 m/s)/ (100 kg+ 15000 kg) v' = (80000 kg-m/s)/ (15100 kg) v' = 5.29 m/s in the direction of the original car's motion

A 40 kg girl skates at 5.5 m/s on ice toward her 70 kg friend who is standing still, with open arms. As they collide and hold each other, what is the speed of the couple? * answer m1v1 + m2v2 = m1v1' + m2v2' v2 = 0 amd v1' = v2' = v' m1v1 = m1v' + m2v' m1v1 = (m1+ m2)v' v' = m1v1 / (m1+ m2) v' = (40 kg)(5.5 m/s)/ (40 kg+ 70 kg) v' = (220 kg-m/s)/ (110 kg) v' = 2.0 m/s in the direction of the original skater's motion

Collisions Type of collision Is Momentum Conserved? Is Energy Conserved? Does it stick? Inelastic Yes No Perfectly Inelastic Elastic

18 Two objects have an elastic collision. Before they collide they are approaching each other with a velocity of 4m/s relative to each other. With what velocity do they go apart from one another? answer 4m/s Initially they are approaching relative to each other at 4m/s, and afterwards they will move apart relative to each other with a velocity of 4m/s. v1+v1'=v2+v2' The difference in the initial velocities is the same as the difference in the final velocities therefore they will move apart relative to one another at the same velcoity