a measure of how hard it is to stop a moving object; it is the product of the object's mass and its velocity; since velocity is a vector, momentum is a vector quantity; symbol is p and SI unit are kg m/sec. The direction of the momentum is the same as that of the velocity.
p= m v
where p is momentum, m is mass in kg, and v is velocity in m/s
High mass objects can have low momentum when they have low velocities; low mass objects can have high momentum when they have high velocities. The more momentum an object has, the harder it is to stop.
Newton's second law of motion expressed in terms of momentum states that the rate of change in momentum of an object is equal to the net force applied to it.
The rate of change of momentum of a body is equal to the net force applied to it.
SF = Dp / D t
An example of momentum change and its magnitude: A ball is thrown at a wall, which stops it. The ball exerts a force on the wall equivalent to its momentum change. If you know the mass of the ball and the original velocity of the ball, you can calculate the momentum change. If you know the length of time it took to stop the ball, you can calculate the force on the ball.
Another example of momentum change and its magnitude: The ball is thrown at the wall and rebounds back toward the thrower. The change in momentum and thus the force is much greater in magnitude. The wall exerts a force to stop the ball but an additional force to give it momentum in the opposite direction. Remember, Dv = vf - vi. In the example described, assume vi = -vf = v, Dv = vf - vi= 2v, not zero.
Impulse
a force exerted over a time interval; symbol is J and SI unit is N sec
J = F t
where J is impulse, F is force, and t is time in seconds
Newton's Cradle demonstration
Collisions and Impulse In a collision of two ordinary objects, both objects are deformed. When the collision occurs, the force jumps from zero at the moment of contact to a very large quantity and back to zero. This occurs over a very brief instant of time. Impulse is useful when dealing with collisions because the forces involved in collisions are usually not constant. It is useful because it lets one calculate the average acceleration experienced by an object that is being acted upon by a non-constant force. The graph below represents force as a function of time during a collision. The curved area of the graph represents Dt, which is usually very small. The yellow area of the graph (the area under the curve) represents the product of force and Dt, or the impulse given to the object. The average force can be estimated from the graph and used with Dt to estimate the impulse. The same area under the curve, or the same impulse, could be obtained by a larger force acting over a shorter time interval or by a smaller force acting over a longer time interval.

According to Newton's second law, an unbalanced force causes a mass to accelerate. Restating Newton's second law in terms of momentum, an impulse causes the velocity of an object with mass to change, therefore causing a change in momentum
J = F t = m Dv = D p
where D stands for "change in"
Impulse explains the operation of air bags, etc. An air bag increases the time that the change in momentum occurs over. The product of the force and Dt must equal the change in momentum. If Dt is increased, then the force must be decreased in order for their product to be constant.
Angular momentum
the product of a rotating object's moment of inertia and its angular velocity; if there is no torque acting on an object, its angular momentum is constant
System
a term that describes a collection of objects
Closed system
mass is constant
Open system
mass is not constant
Isolated system
one in which no external force acts
Law of conservation of momentum
the momentum of a closed, isolated system is constant; the sum of the initial momentum of the objects is equal to the sum of the final momentum of the objects
Spi = S pf
where pi is the initial momentum and pf is the final momentum
Objects transfer their momentum in collisions. The total momentum before the collision is equal to the total momentum after the collision in a closed, isolated system. If one object loses momentum in a collision, then another object must gain that amount of momentum.
The law of conservation of energy previously studied is one of the conservation laws of physics. The law of conservation of momentum introduces another quantitity that is conserved in physics, linear momentum. Other quantities found to be conserved are angular momentum and electric charge.
There are two types of collisions:

• Elastic collision

total momentum and total energy is conserved; elastic collisions only occur on the sub-atomic particle level
An Advanced look at Elastic Collisions  Assume two objects with mass m1 and m2 moving with intial speeds v1 and v2, respectively, collide head on in a perfectly elastic collision. Both momentum and kinetic energy are conserved in elastic collisions. Their final speeds can be written as v1f and v2f, respectively. If the objects have the same mass, they simply exchange velocities in a perfectly elastic collision.
The expression for conservation of linear momentum can be written as:
m1v1 + m2v2 = m1v1f + m2v2f
The expression for conservation of kinetic energy can be written as:
1/2m1v12 + 1/2m2v22 = 1/2m1v1f2 + 1/2m2v2f2
Graphing calculator solution for elastic collisions

• Inelastic collision

total momentum is conserved; real-life collisions are all inelastic.
An Advanced Look at Inelastic Collisions  If two objects stick together after the collision, it is said to be a totally inelastic collision. In inelastic collisions, some of the kinetic energy is converted into other types of energy such as thermal energy or potential energy.The sum of the final kinetic energies is always less than the sum of the initial kinetic energies in an inelastic collision.
Elastic and Inelastic Collisions Interactive Demonstration
Newton's Cradle
One-Dimensional Collision Applets
Interactive Two-Dimensional Collision Applet
Interactive linear collision
Interactive Conservation of Linear Momentum Applet
Linear momentum
objects collide in straight-line motion. The collision occurs in a line, or one dimension.
Method of working linear momentum problems:

• Find the initial momentum of each object. Remember that momentum is a vector quantity--object's velocities are positive or negative. The total initial momentum is the sum of all the object's initial momentum.
• Find the final momentum of each object. Remember that momentum is a vector quantity--object's velocities are positive or negative. The total final momentum is the sum of all the object's initial momentum
• Set the total initial momentum equal to the total final momentum.

Be careful when working conservation of momentum problems! The problems can be algebraically correct, but not be correct according to the law of conservation of physics. Your physics must be correct--not just your algebra!
Momentum in two dimensions
Momentum is conserved in the x-direction and in the y-direction.
Method of working two-dimensional momentum problems:
Interactive 2-D collisions

• When the velocity of the object is at an angle, resolve this resultant velocity into its x- and its y-components. Remember to assign a positive or a negative sign to the components' velocities.
• Conserve the momentum in the x-direction. Find the initial momentum of each object in the x-direction. In other words, only consider the x-component of its velocity. Add the initial momentum of each object to find the total initial momentum. Find the final momentum of each object in the x-direction. In other words, only consider the x-component of its velocity. Add the final momentum of each object to find the total final momentum. Set the total initial momentum in the x-direction equal to the total final momentum in the x-direction.
• Conserve the momentum in the y-direction. Find the initial momentum of each object in the y-direction. In other words, only consider the y-component of its velocity. Add the initial momentum of each object to find the total initial momentum. Find the final momentum of each object in the y-direction. In other words, only consider the y-component of its velocity. Add the final momentum of each object to find the total final momentum. Set the total initial momentum in the y-direction equal to the total final momentum in the y-direction.
• Step two yields the x-component of the desired velocity. Step three yields the y-component of the desired velocity. Use vector addition of the two components to find the resultant velocity of the object.

AP Multiple Choice Questions for Momentum

• Be prepared to perform simple calculations for linear momentum problems.
• Remember signs for your velocities! For example, remember sign conventions for problems in which a ball rebounds from a bat, etc. The sign of its incoming velocity is opposite that of its outgoing velocity. This does not give you a net momentum change of zero!
• Remember that momentum is conserved (the same) in any collision.
• Be able to identify an elastic collision (objects rebound, do not stick together).
• Remember that momentum is conserved, but kinetic energy is not, in inelastic collisions (which represent everything in real life).
• Be able to recognize when momentum is changing from a graph. Momentum changes when speed changes.
• Be able to predict the speeds and directions of objects after a collision.

AP Free Response Choice Questions for Momentum

• Be able to perform calculations for ballistic pendulum problems. These are problems where an object is suspended in such a way that it can swing upward. It is hit with a moving object that becomes imbedded in the suspended object. This collision is the conservation of momentum part of the problem. You can use conservation of momentum to predict the speed of the suspended object/moving object (sometimes you work this problem "backwards" and use conservation of momentum to predict the speed of the moving object). The suspended object/moving object now has kinetic energy. It swings upward a height h, converting its kinetic energy into potential energy. You can use conservation of energy to predict how high it will swing upward.
• Remember that energy lost in an inelastic collision usually results in a thermal energy gain (because of work done against friction).
• Momentum is usually combined with kinetic energy calculations in free response questions.
• Be able to perform calculations for objects attached to a spring. A moving object collides with the object on a spring and becomes stuck to it. The spring is compressed a distance d before it comes to rest. You can use conservation of momentum to predict the initial speed of the moving object/object attached to a spring. Its kinetic energy is now converted into elastic potential energy of the spring. Another variation -- collisions like this usually set the object into SHM, so they can ask you about the period of oscillation of the objects on the spring.
• Use a graph or use data to estimate the average force acting on an object during a collision. Use this to calculate the impulse given to the object and its average acceleration. Remember, impulse better describes a collision where the force is variable.
• You may be asked to work an inelastic collision problem, writing expressions for velocity. Then, they will ask you to do the same problem for an elastic collision!
• Another calculation that is given that combines momentum and motion involves collisions occurring on rough surfaces. Remember that the fricitonal force serves as the unbalanced force that eventually causes the object to stop. You can predict how far it will go before it stops using the work/energy theorem or calculation the frictional force and using acceleration formulas and second law.
• Not very common, but they have asked you to write velocity expressions before and after the collision for momentum in two dimension problems.

Linear Momentum Sample Problems

1. A 4 N force acts on a 3 kg object moving at 8 m/s for 10 sec. What is the object's change in momentum? What impulse acts on the object? What is the object's final speed?
2. A 1000 kg car traveling at 9 m/s, east, strikes a stationary 2000 kg truck. They interlock as a result of the collision and move off as one. What is their speed? What is their velocity?
3. A 15,000 kg rocket launcher holds a 5000 kg rocket. The rocket exits the launcher at +450 m/s. What is the recoil velocity of the launcher?
4. A 100 g ball traveling to the right at 2 m/s strikes a 200 g ball traveling to the left at 4 m/s. After the collision, the 100 g ball has a velocity of 8 m/s, left. What is the velocity of the 200 g ball?
5. A 1200 kg car moving at 8m/s, north, strikes a 2000 kg truck moving at 4 m/s, south. The velocity of the car is 6 m/s, south. What is the velocity of the truck?

Momentum in Two Dimensions - Sample Problems

1. A 1325 kg car traveling north at 27 m/s collides with a 2165 kg car moving east at 17 m/s. As a result of the collision, they stick together. What is their velocity after the collision?
2. A sticky ball with a mass of 200 g is moving to the west at 6 m/s. It collides with another sticky ball with a mass of 300 g moving north at 5 m/s. The sticky balls stick together as a result of the collision and move off as one. What is their velocity?
3. A 6 kg object A moving at 3 m/s, right, collides with a 6 kg object B at rest. After the collision, A moves at 1.6 m/s, 30°. What is the velocity of B after the collision?
4. A stationary 0.14 kg ball is struck by a 0.23 kg ball moving east at 2 m/s. After the collision, the 0.14 kg ball has a velocity of 0.9 m/s, 30°. What is the velocity of the 0.23 kg ball?
5. A 0.50 kg ball at rest is struck by a 0.30 kg ball moving west at 5 m/s. After the collision, the 0.30 kg ball has a velocity of 3 m/s, 200°. What is the velocity of the 0.50 kg ball?
6. A 2 kg object moves at 4 m/s, south. It strikes a 3 kg object at rest. After the collision, the 2 kg object has a velocity of 2.5 m/s, 300°. What is the velocity of the 3 kg object?

Linear Momentum & Impulse Homework

1. An 8 N force acts on a 5 kg object for 3 sec. What impulse is given the object? What change in momentum does this impulse cause? If the object’s initial velocity was 25 m/s. what is its final velocity? Ans: 24 Nsec; 24 Nsec; 29.8 m/s
2. A 6 N force acts on a 3 kg object for 10 sec. What will be the final velocity of the object if its initial velocity was 10 m/s? Ans: 30 m/s
3. A 20,000 N truck is acted upon by a force that decreases its speed from 10 m/s to 5 m/s in 5 sec. What is the magnitude of the force? Ans: -2040.82 N
4. A 2000 N force acts on a rocket of mass 1000 kg, increasing its speed from rest to 200 m/s. How long did the force act? Ans: 100 sec
5. A 700 kg car moving at 20 m/s collides with a stationary truck with mass 1400 kg. The two vehicles interlock as a result of the collision. What is the velocity of the car/truck? Ans: 6.67 m/s
6. A rocket sits on a launch pad loaded with fuel. The igniter causes the rocket to rise off the pad, oxidizing 100 g of fuel and ejecting it out the back of the rocket at -650 m/s. After launch, the rocket has a mass of 3 kg. What is its speed after launch? Ans: +21.67 m/s
7. A 15 g pellet moving at 600 m/s hits a stationary 100 g block and becomes embedded in the block. With what speed do they move off together? Ans: 78.26 m/s
8. A 0.5 kg ball traveling at 6 m/s collides head-on with a 1 kg ball moving in the opposite direction at 12 m/s. The 0.5 kg ball moves away at 14 m/s in the opposite direction after the collision. Find the velocity of the 1 kg ball after the collision. Ans: -2 m/s
9. A plastic ball of mass 0.2 kg moves at 0.6 m/s, east. It collides with a 0.1 kg ball moving at 0.8 m/s, west. After the collision, the velocity of the 0.1 kg ball is 0.6 m/s, east. What is the velocity of the 0.2 kg ball after the collision? Ans: 0.1 m/s, west
10. A 25 g ball moving at 10 m/s, right, collides head-on with a 50 g ball moving at 3 m/s, left. After the collision, the 25 g ball moves at 6 m/s, left. What is the velocity of the 50 g ball after the collision? Ans: 5 m/s, right
11. A 3 kg can containing a 100 g firecracker is thrown into the air. The firecracker explodes, exiting the can at 25 m/s. What is the speed of the can after the collision? Ans: -0.83 m/s
12. A thread holds two carts together that are separated by a compressed spring. The thread is broken. One cart, with mass of 1.5 kg, moves with a velocity of 27 cm/s to the left. What is the velocity of the other cart with mass of 4.5 kg? Ans: 0.09 m/s, right

Momentum in Two Dimensions Homework

1. Hockey puck A moves to the right at 50 m/s. It strikes an identical hockey puck B that is stationary on the ice. After the collision, the velocity of A is 35 m/s, 27.6°. What is the velocity of B? Ans: 24.97 m/s, 40.52° below the horizontal
2. A 600 g billiard ball moving to the right at 2 m/s collides with an 800 g ball at rest. After the collision, the 600 g ball is deflected at an angle of 37° above its original direction at a rate of 0.5 m/s. What is the magnitude and direction of the 800 g ball’s velocity? Ans: 1.22 m/s, 10.85° below the horizontal
3. A 6000 kg truck traveling north at 5 m/s collides with a 4000 kg car traveling west at 15 m/s. The two remain locked together after the collision. What is their velocity after the collision? Ans: 6.71 m/s, 26.6° north of west
4. A 1200 kg car traveling east at 60 km/h collides with a 3000 kg truck traveling north at 40 km/h. After the collision, they remain joined. What is their velocity? Answer in km/h. Ans: 33.3 km/h, 59°
5. A 10 kg ball travels west at 4 m/s. It strikes a 12 kg ball at rest. After the collision, the velocity of the 10 kg ball is 2.5 m/s, 40° below the horizontal. What is the 12 kg ball’s velocity? Ans: 2.20 m/s, 142.4°

AP Objectives - Momentum

• Stuents should understnad impulse and linear momentum so they can:
1. Relate mass, velocity, and linear momentum for a moving body, and calculate the total linear momentum of a system of bodies.
2. Relate impulse to the change in linear momentum and the average force acting on a body.
• Students should understand linear momentum conservation so they can:
1. Identify situations in which linear momentum, or a component of the linear momentum, is conserved.
2. Apply linear momentum conservation to determine the final velocity when two bodies that are moving along the same line, or at right angles, collide and stick together, and calculate how much kinetic energy is lost in such a situation.
3. Analyze collisions of particles in one or two dimensions to determine unknown masses or velocities, and calculate how much kinetic energy is lost in a collision.

AP Momentum Sample Problems
Use a value of 10 m/s2 for g

1. A graphical representation of how force varies with time of a collision of an object with mass of 3 kg with a wall is given above.
• Calculate the impulse given to the object. Ans: 7.5 Nsec
• If the object had an original velocity of +2 m/s, calculate its final velocity. Ans: +4.5 m/s
• If the object had an original velocity of -2 m/s, calculate its final velocity. Ans: +0.5 m/s
• Calculate how much impulse would be required to give the object moving at -2 m/s a final velocity of +4.5 m/s. Ans: 19.5 Nsec
• Explain why it takes for much more impulse to give the -2 m/s object a final velocity of +4.5 m/s than it did to give the +2 m/s object a final velocity of +4.5 m/s.
2. A tennis ball of mass 0.060 kg and speed 25 m/s strikes a wall at an angle of 45° and rebounds with the same speed. What impulse is given the wall? Ans: the absolute value of 2.12 N sec
3. An 18 g rifle bullet traveling at 230 m/s buries itself in a 3.6 kg block hanging on a 2.8 m long string which makes the block swing upward in an arc.
• What is the initial momentum of the bullet? Ans: 4.14 kgm/s
• What is the initial kinetic energy of the bullet? Ans: 476.1 J
• What is the speed of the bullet/block the instant after the collision? Ans: 1.14 m/s
• What is the kinetic energy of the bullet/block the instant after the collision? Ans: 2.37 J
• What percentage of energy was lost in the collision? (Hint: work this as "part over whole.") Ans: 99.5%
• Give one example where the energy was lost in the collision.
• How high will the bullet/block swing? Ans: 0.065 m
4. A 15 g bullet strikes and becomes embedded in a 1.10 kg block of wood placed on a horizontal surface in front of the gun. The coefficient of kinetic friction between the block and surface is 0.25; the impact drives the block 9.5 m before it comes to rest.
• What work is done by the frictional force to bring the bullet/block to rest? Ans: 26.48 J
• Calculate the speed of the bullet/block the instant after the collision. Ans: 6.89 m/s
• Calculate the initial speed of the bullet. Ans: 512.31 m/s
5. A 5800 kg railroad car coasts along with a constant speed of 8.60 m/s on a level track. Snow begins to fall vertically and fills the car at the rate of 3.50 kg/min. Ignoring friction, what is the speed of the car after 90 min? Ans: 8.16 m/s
6. A ball of mass 0.44 kg moving east at 3.70 m/s collides head-on with a 0.220 kg ball at rest. The collision is perfectly elastic. What will be the speed and direction of each ball after the collision? Ans: 1.23 m/s; 4.94 m/s