General Physics I: Day 18 Impulse & The Conservation of Momentum

A 350 kg roller-coaster cart must make it up a hill that is 8.0 m high. If 20,000 J of energy will be “lost” to friction during the climb up the hill, what is the minimum speed it will need when it starts up the hill? v = 16.5 m/s v = 26.5 m/s v = 36.5 m/s v = 46.5 m/s Wnc = Ef – E0 -20,000 = mgh-1/2mv2… so v2 = 2(mgh+20,000)/m v = 16.5 m/s

A very special basketball, dropped (from rest) from a height of 1 meter strikes the earth and returns to a height of 1 meter. For which system is energy conserved during every moment of the process? the basketball by itself the earth by itself the basketball plus the earth all of the above

A Conserved Quantity Working with Newton’s 2nd and 3rd laws we find an interesting quantity. As with energy, if we choose our system well, this new quantity should also be conserved. We name this quantity linear momentum. Or… just momentum: One crucial point: momentum is a vector quantity!

Warm-Up: Momentum vs. KE Describe two significant differences between kinetic energy and momentum. ~58% → Formulas are different (𝑣 vs. 𝑣2) ~75% → Momentum is a vector K is a scalar ~67% → Momentum ↔ 𝐹 & time, 𝐾 ↔ 𝐹 & dist. ~0% → Kinetic energy can become other forms of energy. Momentum is momentum. (or… 𝑝 is conserved, 𝐾 is not) ~0% → False difference of some kind From internet: “Kinetic energy is a scalar, momentum is a vector. Kinetic energy is how much energy a moving body can carry. Momentum determines how that energy is delivered.”

Warm-Up: Momentum vs. KE “Momentum=m*v, making momentum a vector KE=1/2mv^2, making KE a scalar Change in momentum (impulse) depends on time. Change in KE (work) depends on distance.” From internet: “Kinetic energy is a scalar, momentum is a vector. Kinetic energy is how much energy a moving body can carry. Momentum determines how that energy is delivered.”

Warm-Up: Momentum vs. KE “The first difference is that changes in momentum depend on the amount of time a force is applied (as determined by impulse); and changes in kinetic energy depend on the distance over which a force is applied (as determined by work done). The other difference is that momentum is a vector quantity, and kinetic energy is a scalar quantity.” From internet: “Kinetic energy is a scalar, momentum is a vector. Kinetic energy is how much energy a moving body can carry. Momentum determines how that energy is delivered.”

The Real Newton’s 2nd Law No we can see how Newton himself wrote it… This also opens the door for handling situations where the mass of an object changes! (but not yet) As we learn about momentum, this form lets us connect forces (old) to momentum (new).

Warm-Up: Pushing Carts Two cars, one four times as heavy as the other, are at rest on a frictionless horizontal track. Equal forces act on each of these cars for a duration of exactly 5 seconds. The momentum of the lighter car will be _______ the momentum of the heavier car. 50% → one-quarter 0% → one-half 22% → equal to 7% → twice 22% → four times Lets hit the important point here…

The Impulse-Momentum Theorem 𝐽 is the impulse. This assumes 𝐹 net is constant. This is the impulse-momentum theorem:

Two cars, one four times as heavy as the other, are at rest on a frictionless horizontal track. Equal forces act on each of these cars for a duration of exactly 5 seconds. The momentum of the lighter car will be _______ the momentum of the heavier car. one-quarter one-half equal to twice four times

Impulse From A Varying Force Lets example impulse from a graphical standpoint. We look at a plot of F vs. t: Impulse ( 𝐽 ) is the area underneath the curve.

Impulse From A Realistic Force

Sample Problem with Help A 145 g baseball is moving horizontally to the right at 40.0 m/s as it passes the batter. The batter hits the ball and the ball goes straight up at 40.0 m/s. In which direction is the average force of the bat directed? A)  B)  C)  D)  NOTE: The last question on this page will appear after you have ended the question and click as if to go to the next slide. You can then ask the follow-up question and show how to find the solution (see below). Look at change in momentum vector. Hypotenuse of triangle with pi and pf as legs is the change in momentum. (mv)2 + (mv)2 = (delta p)2, so delta p = 8.2 kg m/s. If delta p = Favg (delta t), then Favg = 1640 N! Interesting facts… the baseball is often compressed to half its diameter. The bat gets compressed by about 2%. Emphasize vector nature of the problem. Any baseball players? Ever get any advice about "swinging for the fences"? It is a bad idea to swing upward (like #2 above)… only leads to pop-flies.

Sample Problem with Help A 145 g baseball is moving horizontally to the right at 40.0 m/s as it passes the batter. The batter hits the ball and the ball goes straight up at 40.0 m/s. If the bat is in contact with the ball for only 5.0 ms what is the average force exerted by the bat? NOTE: The last question on this page will appear after you have ended the question and click as if to go to the next slide. You can then ask the follow-up question and show how to find the solution (see below). Look at change in momentum vector. Hypotenuse of triangle with pi and pf as legs is the change in momentum. (mv)2 + (mv)2 = (delta p)2, so delta p = 8.2 kg m/s. If delta p = Favg (delta t), then Favg = 1640 N! Interesting facts… the baseball is often compressed to half its diameter. The bat gets compressed by about 2%. Emphasize vector nature of the problem. Any baseball players? Ever get any advice about "swinging for the fences"? It is a bad idea to swing upward (like #2 above)… only leads to pop-flies.

Considering Systems Using Newton’s laws (linear or rotational) it is usually clear what objects are of concern to us: We choose one object to focus on (at a time) We look at everything that interacts with it When we use conservation laws, we must carefully Choose A System: We (almost) never choose one object We have to include all objects that are “involved”

Newton’s Laws for a System Newton's laws describe the motion of the system: Why do we only consider external forces? All internal forces come in pairs (of course)… so they cancel out! For a system we can rewrite this with momentum:

Conservation of Momentum Consider this carefully… If there is no net external force on a system, the total momentum is conserved. Once more: For a system with no net external force: Careful: The momentum of individual objects may (will) change, only the total is constant.

Difficult/Interesting “What I thought was interesting was that the Law of Conservation of Momentum is more general than the Law of Conservation of Energy. It interesting that energy doesn't necessarily need to be conserved in order for momentum to be conserved.”

Worked-Example: 1D Collision What is 𝑣 1 𝑓𝑖𝑛𝑎𝑙?

Worked-Example: 1D Collision

Worked-Example: 1D Collision

Applying Conservation of Momentum As with kinematics, conservation laws require that we decide on an initial and a final “state”. If a net external force exists, we can still proceed with conservation of momentum if we consider a very short time-span: Long enough to cover the action Short enough to keep external forces from changing the total momentum. (Called the impulse approximation)

Applying Conservation of Momentum As with kinematics, conservation laws require that we decide on an initial and a final “state”. If a net external force exists, we can still proceed with conservation of momentum if we consider a very short time-span (impulse approximation). Determine knowns/unknowns about initial/final. Split momentums into components! Apply the conservation of momentum on each axis:

Sample Problem A basketball player (60.0 kg) needs to save a basketball (0.600 kg) that is going out of bounds. She leaps into the air and grabs the ball. At this point she and the ball are moving forward at 2.00 m/s. She then throws the ball backward at 10.0 m/s (relative to the floor). How fast is she moving (horizontally) when she crashes into the stands?

Suppose you are on a cart, initially at rest on a track with very little friction. You throw balls at a partition that is rigidly mounted on the cart. If the balls bounce straight back as shown in the figure, is the cart put in motion? Yes, to the right. Yes, to the left. No, it stays in place.

Coming up… Thursday (10/23) → 8.3 – 8.6 Ch. 8 Homework due Sunday by 11:59 PM Warm-Up due Wednesday by 10:00 PM