Topic 2: Mechanics 2.4 Momentum and impulse Essential idea: Conservation of momentum is an example of a law that is never violated. Nature of science: The concept of momentum and the principle of momentum conservation can be used to analyse and predict the outcome of a wide range of physical interactions, from macroscopic motion to microscopic collisions. Topic 2: Mechanics 2.4 Momentum and impulse Understandings: Newtons second law expressed in terms of rate of change of momentum Impulse and force time graphs Conservation of linear momentum Elastic collisions, inelastic collisions and explosions

Topic 2: Mechanics 2.4 Momentum and impulse Applications and skills: Applying conservation of momentum in simple isolated systems including (but not limited to) collisions, explosions, or water jets Using Newtons second law quantitatively and qualitatively in cases where mass is not constant Sketching and interpreting force time graphs Determining impulse in various contexts including (but not limited to) car safety and sports Qualitatively and quantitatively comparing situations involving elastic collisions, inelastic collisions and explosions Topic 2: Mechanics 2.4 Momentum and impulse

Guidance: Students should be aware that F = ma is the equivalent of F = p / t only when mass is constant Solving simultaneous equations involving conservation of momentum and energy in collisions will not be required Calculations relating to collisions and explosions will be restricted to one-dimensional situations A comparison between energy involved in inelastic collisions (in which kinetic energy is not conserved) and the conservation of (total) energy should be made Topic 2: Mechanics 2.4 Momentum and impulse Data booklet reference:

Impulse Topic 2: Mechanics 2.4 Momentum and impulse International-mindedness: Automobile passive safety standards have been adopted across the globe based on research conducted in many countries Theory of knowledge: Do conservation laws restrict or enable further development in physics? Utilization: Jet engines and rockets Martial arts Particle theory and collisions (see Physics sub-topic 3.1)

Topic 2: Mechanics 2.4 Momentum and impulse Aims: Aim 3: conservation laws in science disciplines have played a major role in outlining the limits within which scientific theories are developed Aim 6: experiments could include (but are not limited to): analysis of collisions with respect to energy transfer; impulse investigations to determine velocity, force, time, or mass; determination of amount of transformed energy in inelastic collisions Aim 7: technology has allowed for more accurate and precise measurements of force and momentum, including video analysis of real-life collisions and modelling/simulations of molecular collisions Topic 2: Mechanics 2.4 Momentum and impulse

Newtons second law in terms of momentum Linear momentum, p, is defined to be the product of an objects mass m with its velocity v. linear momentum Its units are obtained directly from the formula and are kg m s-1. EXAMPLE: What is the linear momentum of a 4.0-gram NATO SS 109 bullet traveling at 950 m/s? SOLUTION: Convert grams to kg (jump 3 decimal places left) to get m = Topic 2: Mechanics 2.4 Momentum and impulse Newtons second law in terms of momentum p = mv linear momentum

This last is Newtons second law in terms of change in momentum rather than mass and acceleration. = Newtons second law (p-form) EXAMPLE: A 6kg object increases its speed from 5 ms-1 to 25 m s-1 in 30 s. What is the net force acting on it? SOLUTION: Topic 2: Mechanics 2.4 Momentum and impulse Kinetic energy in terms of momentum =

linear momentum = 1 2 2 kinetic energy EXAMPLE: Show that kinetic energy can be calculated directly from the momentum using the following: 2 kinetic energy = 2 SOLUTION: From we obtain . Then

Topic 2: Mechanics 2.4 Momentum and impulse Kinetic energy in terms of momentum kinetic energy PRACTICE: What is the kinetic energy of a 4.0-gram NATO SS 109 bullet traveling at 950 m/s and having a momentum of 3.8 kg m s-1? SOLUTION: Start from scratch using or you can use Topic 2: Mechanics 2.4 Momentum and impulse Collisions A collision is an event in which a relatively strong force acts on two or more bodies for a relatively short time.

The Meteor Crater in the state of Arizona was the first crater to be identified as an impact crater. Between 20,000 to 50,000 years ago, a small asteroid about 80 feet in diameter impacted the Earth and formed the crater. Topic 2: Mechanics 2.4 Momentum and impulse Collisions Consider two colliding pool balls system boundary

system boundary system boundary FYI A __________ ___________ is the area of interest used by physicists in Before the study of phase complex During processes. A _________ phase

_________ has After no work done phase on its parts by external forces. Topic 2: Mechanics 2.4 Momentum and impulse Collisions If we take a close-up look at a collision between two bodies, we can plot the force acting on each mass during the collision vs. the time : vAi vBi Before F B A During phase

Before After FAB FBA A B t During FAB A B FBA phase FAB FBA FYI A B Note the perfect vBf After symmetry of the actionvAf B A phase reaction force pairs.

Topic 2: Mechanics 2.4 Momentum and impulse Force Impulse and force time graphs Although the force varies with time, we can simplify it by averaging it out as follows: Imagine an ant farm (two t sheets of glass with sand in between) filled with the sand in the shape of the above force curve: We now let the sand level itself out (by tapping or shaking the ant farm): The area of the rectangle is the same as the area under the original force vs. time curve.

The ______________ F is the height of this rectangle. F t Force Topic 2: Mechanics 2.4 Momentum and impulse Force Impulse and force time graphs We define a new quantity called ___________ J as the average force times the time. This amounts to the area under the force vs. time graph.

F t t t impulse Since we see that and so we can interpret the impulse as the change in momentum of the object during the collision. impulse Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs J = F t = p = area under F vs. t graph impulse

It is well to point out here that during a collision there F t are two objects interacting with one another. Because of Newtons third F law, the forces are equal but opposite so that F = - F. Thus for one object, the area (impulse or momentum change) is positive, while for the other object the area (impulse or momentum change) is negative. FYI Thus impulse can be positive or negative. Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs EXAMPLE: A 0.140-kg baseball comes in at 40.0 m/s, strikes the bat, and goes back out at 50.0 m/s. If the

collision lasts 1.20 ms (a typical value), find the impulse imparted to the ball from the bat during the collision. SOLUTION: v0 = p0 = Before p0 = We can use J = p: vf = J = pf p0 pf = = After pf = = FYI The units for impulse can also be kg m s-1.

Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs EXAMPLE: A 0.140-kg baseball comes in at 40.0 m/s, strikes the bat, and goes back out at 50.0 m/s. If the collision lasts 1.20 ms (a typical value), find the average force exerted on the ball during the collision. SOLUTION: We can use . Thus Fmax F FYI Fmax is even greater than F! Topic 2: Mechanics 2.4 Momentum and impulse Sketching and interpreting force time graphs J = F t = p = area under F vs. t graph

impulse Force F / n PRACTICE: A bat striking a ball imparts a force to it as shown in the graph. Find the impulse. SOLUTION: Break the graph into simple areas of rectangles and triangles. 9 6 3 0 0 5 Time t / s

10 Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs EXAMPLE: T v How does a jet engine produce thrust? SOLUTION: u The jet engine sucks in air (at about the speed that the plane is flying through the air), heats it up, and expels it at a greater velocity. The momentum of the air changes since its velocity

does, and hence an impulse has been imparted to it by the engine. The engine feels an equal and opposite impulse. Hence the engine creates a thrust. Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs EXAMPLE: Show that . SOLUTION: From we have This is a 2stage rocket. The orange tanks hold fuel, and the blue tanks hold

oxidizer. The oxidizer is needed so that the rocket works without air. FYI The equation is known as the rocket engine equation because it shows us how to calculate the thrust of a rocket engine. The second example will show how this is done. Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs T EXAMPLE: What is the purpose of the rocket nozzle?

SOLUTION: In the combustion chamber the gas particles have random directions. The shape of the nozzle is such that the particles in the sphere of combustion are deflected in such a way that they all come out antiparallel to the rocket. This ______________________________. The rocket feels ______________________________ ____________________________________________. Topic 2: Mechanics 2.4 Momentum and impulse Impulse and force time graphs =( rocket engine equation

) EXAMPLE: A rocket engine consumes fuel and oxidizer at a rate of 275 kg s-1 and used a chemical reaction that gives the product gas particles an average speed of 1250 ms-1. Find the thrust produced by this engine. SOLUTION: The units of are kg s-1 so that clearly = The speed v = 1250 ms-1 is given. Thus Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum Recall Newtons second law (p-form): Newtons second law (p-form) If the net force acting on an object is zero, we have

In words, if the net force is zero, then the momentum does not change p is constant. conservation of linear momentum FYI If during a process a physical quantity does not change, that quantity is said to be conserved. Topic 2: Mechanics 2.4 Momentum and impulse The internal forces cancel Conservation of linear momentum Recall that a system is a collection of more than one

body, mutually interacting with each other for example, colliding billiard balls: Note that Fnet = Fexternal + Finternal. But Newtons third law guarantees that Finternal = 0. Thus we can refine the conservation of momentum: conservation of If Fext = 0 then p = CONST linear momentum Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum If Fext = 0 then p = CONST conservation of linear momentum

EXAMPLE: A 2500-kg gondola car traveling at 3.0 ms-1 has 1500-kg of sand dropped into it as it travels by. Find the initial momentum of the system. SOLUTION: The system consists of sand and car: p0,car = p0,sand = p = Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum If Fext = 0 then p = CONST conservation of linear momentum

EXAMPLE: A 2500-kg gondola car traveling at 3.0 ms-1 has 1500-kg of sand dropped into it as it travels by. Find the final speed of the system. SOLUTION: The initial and final momentums are equal: p0 = pf = Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum If Fext = 0 then p = CONST conservation of linear momentum

EXAMPLE: A 12-kg block of ice is struck by a hammer so that it breaks into two pieces. The 4.0-kg piece travels travels at +16 m s-1 in the x-direction. What is the velocity of the other piece? 8 4 SOLUTION: Make before/after sketches! The initial momentum of the two is 0. 4 16 8 v From p = CONST we have p0 = pf. Since p = mv, we see that Topic 2: Mechanics 2.4 Momentum and impulse before 25

0 730 1800 after 730 +1800 Conservation of linear momentum If Fext = 0 then p = CONST conservation of linear momentum EXAMPLE: A 730-kg Smart Car traveling at 25 m s-1 (xdir) collides with a stationary 1800-kg Dodge Charger. The two vehicles stick together. Find their velocity immediately after the collision.

SOLUTION: Make sketches! vf Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum If Fext = 0 then p = CONST EXAMPLE: A loaded Glock-22, having a mass of 975 g, fires a 9.15-g bullet with a muzzle velocity of 300 ms-1. Find the guns recoil velocity. SOLUTION: Use p0 = pf. Then conservation of linear momentum

Topic 2: Mechanics 2.4 Momentum and impulse Conservation of linear momentum If Fext = 0 then p = CONST conservation of linear momentum EXAMPLE: A loaded Glock-22, having a mass of 975 g, fires a 9.15-g bullet with a muzzle velocity of 300 ms-1. Find the change in kinetic energy of the gun/bullet system. SOLUTION: Use EK = mv 2 so EK0 = 0 J. Then Topic 2: Mechanics 2.4 Momentum and impulse

Comparing elastic collisions and inelastic collisions In an ________________, ______________________ (it does not change). Thus __________________. EXAMPLE: Two billiard balls colliding in such a way that the speeds of the balls in the system remain unchanged. The red ball has the same speed as the white ball Both balls have same speeds both before and after Topic 2: Mechanics 2.4 Momentum and impulse Comparing elastic collisions and inelastic collisions In an ___________________, ___________________ ____________ (it does change). Thus _________. EXAMPLE: A baseball and a hard wall colliding in such a way that

the speed of the ball changes. Topic 2: Mechanics 2.4 Momentum and impulse Comparing elastic collisions and inelastic collisions In a _____________________________ the colliding bodies stick together and end up with the same velocities, but different from the originals. __________. EXAMPLE: Two objects colliding and sticking together. u1 v u2 v

The train cars hitch and move as one body The cars collide and move (at first) as one body Topic 2: Mechanics 2.4 Momentum and impulse Comparing elastic collisions and inelastic collisions An explosion is similar to a completely inelastic collision in that the bodies were originally stuck together and began with the same velocities. ___________. EXAMPLE: Objects at rest suddenly separating into two pieces. A block of ice broken in two by a hammer stroke A bullet leaving a gun Topic 2: Mechanics 2.4 Momentum and impulse

Quantitatively analysing inelastic collisions conservation of If Fext = 0 then p = CONST linear momentum EXAMPLE: Two train cars having equal masses of 750 kg and velocities u1 = 10. m s-1 and u2 = 5.0 m s-1 collide and hitch together. What is their final speed? u1 v u2 v SOLUTION: Use momentum conservation p0 = pf. Then Topic 2: Mechanics

2.4 Momentum and impulse Quantitatively analysing inelastic collisions conservation of If Fext = 0 then p = CONST linear momentum EXAMPLE: Two train cars having equal masses of 750 kg and velocities u1 = 10. m s-1 and u2 = 5.0 m s-1 collide and hitch together. Find the change in kinetic energy. u1 v u2 v SOLUTION: Use EK = mv 2. Then

Topic 2: Mechanics 2.4 Momentum and impulse Quantitatively analysing inelastic collisions conservation of If Fext = 0 then p = CONST linear momentum EXAMPLE: Two train cars having equal masses of 750 kg and velocities u1 = 10. m s-1 and u2 = 5.0 m s-1 collide and hitch together. Determine the type of collision. u1 SOLUTION: v u2 v

Topic 2: Mechanics 2.4 Momentum and impulse Quantitatively analysing inelastic collisions conservation of If Fext = 0 then p = CONST linear momentum EXAMPLE: Two train cars having equal masses of 750 kg and velocities u1 = 10. m s-1 and u2 = 5.0 m s-1 collide and hitch together. Was mechanical energy conserved? u1 SOLUTION: v u2 v

Topic 2: Mechanics 2.4 Momentum and impulse Quantitatively analysing inelastic collisions conservation of If Fext = 0 then p = CONST linear momentum EXAMPLE: Two train cars having equal masses of 750 kg and velocities u1 = 10. m s-1 and u2 = 5.0 m s-1 collide and hitch together. Was total energy conserved? u1 SOLUTION: v u2 v

Topic 2: Mechanics 2.4 Momentum and impulse Quantitatively analysing inelastic collisions EXAMPLE: Suppose a .020-kg bullet traveling horizontally at 300. m/s strikes a 4.0-kg block of wood resting on a wood floor. How fast is the block/bullet combo moving immediately after collision? SOLUTION: If we consider the bullet-block combo as our system, there are no external forces in the x-direction at collision. f Topic 2: Mechanics 2.4 Momentum and impulse s

Quantitatively analysing inelastic collisions EXAMPLE: Suppose a .020-kg bullet traveling horizontally at 300. m/s strikes a 4.0-kg block of wood resting on a wood floor. The block/bullet combo slides 6 m before coming to a stop. Find the friction f between the block and the floor. SOLUTION: Use the work-kinetic energy theorem: EK = W f Topic 2: Mechanics 2.4 Momentum and impulse s Quantitatively analysing inelastic collisions EXAMPLE: Suppose a .020-kg bullet traveling horizontally at 300. m/s strikes a 4.0-kg block of wood

resting on a wood floor. The block/bullet combo slides 6 m before coming to a stop. Find the dynamic friction coefficient d between the block and the floor. SOLUTION: Use f = dR: Make a free-body diagram to f find R: F s Topic 2: Mechanics 2.4 Momentum and impulse Quantitatively analysing inelastic collisions EXAMPLE: Suppose a .020-kg bullet traveling horizontally at 300. m/s strikes a 4.0-kg block of wood resting on a wood floor. If the bullet penetrates .060 m of the block, find the average force F acting on it during its collision. SOLUTION: Use the work-kinetic energy theorem on only the bullet: