Understanding Momentum: Definition, Equations, and Applications

1. Momentum Can be defined as inertia in motion (or by Newton: an object’s quantity of motion) Equation: p = m v units: kg m/s So if the object’s at rest, then its p = 0 no matter how massive it is. Since momentum is based on an object’s velocity, which is a vector quantity, it too is an vector quantity – direction matters again!

2. Change in Momentum Δp ≠ p of an object! Δp = pf – pi where each term can have different values. Since Δp most commonly from a Δv, Δp = mvf – mvi Δp= m (vf – vi ) [= mΔv, but don’t use that in math sol’ns!] But it can also be from a Δm Ex: full vs empty salt truck rocket ship burning fuel

3. True or False? If an object’s Δp = 0, then its p = 0 as well. False Ex: Any moving object with a constant velocity If an object’s p = 0, then its Δp = 0 as well. also False, although not as often Only if p = 0 only lasts for an instant during the time when we’re determining Δp Ex: a ball at the top of a free fall climb

4. The Cause of a Change in Momentum Δp is caused by a net force applied for a period of time This is called impulse (J) equations: J = Δp = ΣFΔt = m (vf – vi) units: N s = kg m/s (kg m/s2) s = kg m/s Notice how net force can be defined in terms of p? ΣF = Δp/Δt or “net F = rate of Δp” This is actually how Newton originally stated his 2nd law in his book, PhilosophiæNaturalis Principia Mathematica aka…

5. Mathematical Principles of Natural Philosophy or often simply referred to as the Principia a work in three books by Isaac Newton first published in Latin in July 1687. Lex II: Mutationemmotusproportionalem essevi motriciimpressae, et fierisecundum lineamrectam qua vis illaimprimitur. Translated by Motte 1729 as: Law II: The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd. So, according to modern ideas of how Newton was using his terminology, Law II: The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.

6. Uses for Change in Momentum Equations The equations: J = Δp = ΣFΔt = m (vf – vi) can certainly be applied to something like a car speeding up or slowing down in traffic. a ball being pulled to the ground by gravity. And we will see them used for these “non-collision” changes in momentum… but we already have other physics that works very well to solve for unknowns in these types of problems: constant acceleration equations Newton’s 2nd Law conservation of mechanical energy But these physics ideas don’t work well for collisions because: force and acceleration are never constant, internal energy forms are always present. So these equations (and eventually the conservation of momentum eq’n) are great for collisions… let’s see:

7. Using Change in Momentum to Study Collisions During a collision, net force can be assumed to be due to just the force between the interacting objects, since: The other forces present, like gravity or friction, are often comparativelysmall to the impact force. That’s the force responsible for the ∆p or J anyway! So ‘ΣF’ in ΣFΔt, can just be F, & often it’s actually avg F… & Δt – the time it takes the collision to take place – is often very short … watch… (baseball into bat) Classic F vsΔt graphs for a typical collision

8. Intentional Collisions An intentional collision is one that you planned or wanted to happen, like Bat against incoming baseball Club against golf ball resting on a tee Hammer against nail into the wall Foot into soccer ball Generally in these situations, you are looking for a good amount of change in momentum to happen – you want the ball’s/nail’s mass to change it’s velocity significantly, so you maximize your force, and well as the time it’s applied (usually by something called “follow through” in sports) to maximize the impulse you impart on the object, according to Δp = m (vf – vi) = J= F Δt, But when the collision is not planned, like a car accident or dropping a plate on the floor, that’s a different story…

9. Unintentional Collisions Δp = m (vf – vi) = J= F Δt, If an object is destined to crash, it’s vi and mass are likely unchangeable at that moment, and it’s vf = 0, so that means Δp and J are a fixed value too. That leaves the only hope to save the object with F∙Δt, where if their product = a fixed value, then they are inversely proportional to each other for that situation. Since it’s actually force that causes damage to things, not Δp or J, we want to minimize that force, so we try to extend the amount of time the Δp (collision) takes place in because, The longer the time an object takes to change momentum, the less force will be needed, therefore the less damage to the object. Ex: hard floor vs carpet

10. More examples: Run-away truck ramps Landing stiff legged vs legs bent. Proper way to catch a water balloon.

11. More examples: Air bags in vehicles

12. The Significance of Bouncing When an object bounces, not only did something have to get the object stopped from its original motion, but then it also needed to get it moving again from rest in the opposite direction This requires a greater Δp or impulse than to simply stop the object from moving – and since a “bounce off” often takes no more time than just stopping, it follows that a larger force is present Examples Pelton paddle wheel Rubber bumpers on cars?? (like those on Grande Prix) karate chop that doesn’t work

13. The Law of Conservation of Momentum The momentum of any isolated system remains constant. system refers to the objects you’ve chosen to be included - if they interact, it’s with INTERNAL forces isolated – a system that has no net force being applied from objects outside the defined system - called EXTERNAL forces 2 Ways Con of p is Easily Seen: if one object loses p, then another in the system must gain = p Ex: billiard balls during a game of pool Newton’s Cradle air track

14. Figure 7-5Connecting Train Cars What can we say about the CG of the system… before? After?

15. Figure 7-34 Bumper Car Rear End Collision CG of the system… before? After? v’A = 3.62 m/s v’B = 4.42 m/s Math is not doable in head, but note: • Slower one gains speed, • Faster one loses speed, • vA - vB = - (v’A- v’B) 0.8 m/s = - (-0.8 m/s) Note… always true for 1D elastic

16. The Law of Conservation of Momentum [The momentum of any isolated system remains constant. 2 Ways Con of p is Easily Seen:] if one object starts moving one way, then another in the system will move the opposite way with = p Ex: 2 students face off on skateboards

17. Example of Conservation of Momentum:Rocket Propulsion Recall by 3rd law, Fr on g = -Fg on r Now if we ·Δt ·Δt Then we get Δpgbcr = -Δpr bc g… con of p! What can we say about the CG of the system… before? After?

18. Example of Conservation of Momentum:Recoil of a Fired Gun CG of the system… before? After?

19. Figure 7-3 Momentum is conserved in a collision of two balls, labeled A and B. Where is the CG of the system… before? After?

20. Forces on the balls during the collision of Fig. 7-3. FBonAFAonB These are the A/R forces from Newton’s 3rd Law • To find the net internal force of an isolated system, they will be added together! • And will, therefore, cancel… • So the ΣFint = 0 for any isolated system

21. Conservation of Momentum When it comes to defining your system, you get to pick the objects included in the system If they’re not in, and they apply a force, it’s an external force, so then it’s no longer an isolated system, so we can’t expect p to be conserved. So then Δpsys ≠ 0, and Impulse (J) = Δpsys = FextΔt = m (vf – vi) Ex: 1. push on car from outside of it 2. drop ball – it accelerates to ground 3. any interaction /problem from 7.1 & 7.3 These are NOT exceptions to the law of conservation of momentum, we just aren’t satisfying the requirements of the law!

22. Conservation of Momentum [When it comes to defining your system, you get to pick the objects included in the system] If they’re in and they apply a force, it’s an internal force, so Δpsys = 0 for the system as a whole, even though objects inside may be changing their individual p’s Ex: 1. 2 students face off on skateboards 2. push on car from inside it 3. push car from outside, but include earth 4. watch ball drop, but include earth Does the Earth really gain = & - momentum? Yes – but too small to measure No – since it probably isn’t even a net force

23. Back to vectors big time!! Use directions! • ID the given carefully & use sketch with dir key • Instead of subscripts 1 & 2, use letters that represent specific objects in the problem • If something starts from or goes to rest, then that entire term = 0 • If the objects are stuck together, then they have = v’s so you can pull it out as a common factor: use (m1+ m2) vi , not m1vi1 + m2vi2 The math for the law of conservation of p: for an isolated system: pisys = pfsys m1vi1 + m2vi2 = m1vf1 + m2vf2 m1 = m2 = vi1 = vi2 = vf1 = vf2 =

24. Types of Interactions • Elastic Collisions – kinetic energy is conserved before and after the collision But not during the collision: KE  PEe, until “minimum separation distance” is reached So then there’s no heat loss, no permanent deformation, no sound created! Ex: atomic & molecular collisions But, in the macro world, only occurs as an ideal… since really there’s always some energy lost to heat, so true elastic collisions don’t really exist – but in Physicsland, we have close approximations: Ex: billiard balls in a game of pool steel balls on Newton’s cradle spring bumpers on gliders on air track

25. Types of Interactions • Inelastic Interactions – where KE is not conserved • Inelastic Collisions: KEi > KEf • Ex: minor car accident, bouncing ball, any hit in sports, fired bullet passes straight thru object • Inelastic Explosions: KEi < KEf • Ex: firecracker, bomb (spring bug toy- not isolated) • Completely Inelastic Collision – where not only is KE not conserved, but the maximum lost during the collision is how much stays lost even after it’s over. When the objects entangle / stick together as a result of the collision – maximum permanent deformation • Ex: serious car accident, fired bullet lodges into object, train cars hooking up (gliders w/ magnets) Note: p is conserved throughout all types of interactions!

26. Figure 7-13Elastic and Inelastic Collisions (e) If completely inelastic, they’d have stuck together – maybe still moving, maybe not

27. The Math of Different Types of Interactions • Any Isolated Interaction (elastic or inelastic) If only 1 unknown – only need 1 eq’n: Use Law of Con of p: pisys = pfsys • Elastic Collisions If 2 unknowns – need 2eq’ns: Use Law of Con of p: pisys = pfsys Use Con of KE: Ki = Kf • Head-on Use Law of Con of p: pisys = pfsys Use: vi1 – vi2 = vf2 – vf1 (see derivation p.176)

28. The Math of Different Types of Interactions • Inelastic Collisions – classic ex: ballistic pendulum Must watch because often outside forces act at least for a portion of the entire situation, and if so, then Δpsys ≠ 0  p is not conserved in that portion • For the collison portion, it’s isolated, so Use Law of Con of p: pisys = pfsys • For the stages where there are external forces, then try to conserve energy In general, use L of C of E: Emechi + WNC = Emechf If external forces are conservative forces, then WNC = 0 If external forces are nonconservative forces, then usually Emechf = 0

29. Ballistic Pendulum Set Up Stage I: isolated system bullet moving toward stationary block Stage II: bullet imbeds in block, moving together still isolated, but total inelastic collision Stage III: gravity transforms all KE into PEg no longer isolated, but only conservative forces acting But what if block was not a pendulum, but attached to a spring that either extended or compressed after catching the bullet? Or slid to rest along a surface after catching the bullet?

30. Momentum Conservation in 2 or 3 Dimensions What can we say about the CG of these systems… Before? After? Yes, it’s conserved in 2/3D just like 1D: pisys= pfsysstill! But now the math requires 2 (or 3) D vector addition: • Create a vector diagram (tip to tail) of the individual momentum vectors of the each objects in the system. • Use vector addition math like right triangle trig or law of sine/cosine to solve. Try sample problem on handout…