1. Topics for Today Lab workbooks are available in the book store
Set clicker to channel 44
Linear momentum (9-3)
Collision and impulse (9-4)
Conservation of linear momentum (9-5)
Momentum and kinetic energy in collisions (9-6)

2. Linear Momentum
Physicists really like conservation laws. Velocities of particles are not conserved, but a related quantity called the “momentum” is conserved.
Linear momentum is 𝑝 =𝑚 𝑣
Newton’s second law is then 𝐹 = 𝑑 𝑝 𝑑𝑡
If we insert the definition of momentum, we get back the 2nd law
𝐹 = 𝑑 𝑝 𝑑𝑡 = 𝑑 𝑑𝑡 𝑚 𝑣 =𝑚 𝑑 𝑣 𝑑𝑡 =𝑚 𝑎
Newton originally wrote his second law in the momentum form.

3. Linear Momentum for a System
For a system of particles, the total linear momentum is
𝑃 = 𝑝 𝑝 𝑝 3 +…+ 𝑝 𝑛 = 𝑚 1 𝑣 𝑚 2 𝑣 2 + 𝑚 3 𝑣 3 +…+ 𝑚 𝑛 𝑣 𝑛
If you compare this to the definition of the center of mass, you find (after taking one time derivative) that 𝑃 =𝑀 𝑣 𝑐𝑚 where M is the total mass of the system.
If we take the time derivative, then we find 2nd law for a system
𝑑 𝑃 𝑑𝑡 =𝑀 𝑑 𝑑𝑡 𝑣 𝑐𝑚 =𝑀 𝑎 𝑐𝑚 = 𝐹 𝑛𝑒𝑡 ⇒ 𝐹 𝑛𝑒𝑡 = 𝑑 𝑃 𝑑𝑡
As before, this looks exactly like the 2nd law for a single particle.

4. Collisions and Impulse
Sometimes, particles interact so fast that we can’t keep up with the details. In these cases, like collisions, we might not be able to find the force as a function of time, but we can find the integral of the force that we call the “impulse”. Integrating the 2nd law w.r.t. time, we find
𝐽 = 𝐹 𝑑𝑡 = 𝑑 𝑝 𝑑𝑡 𝑑𝑡= 𝑝 𝑓 − 𝑝 𝑖 =∆ 𝑝
Quiz – a baseball with mass of 150 grams and a horizontal speed of 40 m/s is hit with a bat. The ball ends up moving horizontally in the opposite direction with a speed of 60 m/s. What impulse did the bat give to the ball? A=15 kg, B=15 m/s, C=15 kg∙m/s, D=15 kg∙m/s2

5. Collisions and Impulse
Usually, the force is some complicated function of time.
In some cases, we know only the average force, 𝐹 𝑎𝑣𝑔 , exerted over some interval of time, ∆𝑡. The impulse is then
𝐽 = 𝐹 𝑑𝑡 = 𝐹 𝑎𝑣𝑔 ∆𝑡
Quiz – The baseball in the previous problem received an impulse of 15 kg∙m/s. If the bat and baseball were in contact for 0.7 milliseconds, what was the average force exerted on the bat by the ball?
A=21 N, B=210 N, C=2100 N, D=21000 N

6. Collisions and Impulse
What is the sign of Δpx? A = positive, B = negative, C = zero
What is the sign of Δpy? A = positive, B = negative, C = zero
What is the direction of ∆ 𝑝 ? A = +x, B = -x, C = +y, D = -y

7. Conservation of Linear Momentum
If we have an isolated system with no external forces acting on it, then
𝐹 𝑛𝑒𝑡 = 𝑑 𝑃 𝑑𝑡 =0 or ∆ 𝑃 =0 or 𝑃 𝑓 = 𝑃 𝑖
Momentum is conserved!
Do clacker demo. Why is the number of balls moving conserved?
Do demo 1N20.20 Conservation of linear momentum – Air track.
Do demo with equal masses.
Quiz – what is the ratio of speeds (in the final state) if the ratio of masses is 1:3? A=1:1, B=1:3, C=3:3, D=3:1
Do demo with 1:3 mass ratio.

8. Conservation of Linear Momentum
Quiz – a car accelerates from rest. In doing so the car’s momentum changes by a certain amount. The Earth’s moment changes by
Zero
The same amount
And equal and opposite amount
The answer depends on the interaction between the two.

9. Momentum and Energy in Collisions
If two (or more) objects collide then 𝑃 𝑓 = 𝑃 𝑖 , momentum is conserved. This is always true as long as there are no external forces.
While energy is also always conserved, this is not necessarily true of kinetic energy. We find two types of collisions:
Elastic collisions = kinetic energy is conserved
Inelastic collisions = kinetic energy is not conserved.
The greatest loss in kinetic energy occurs when two objects collide and then stick together. This is a “completely inelastic collision”.

10. Inelastic Collisions
If two (or more) objects collide then 𝑃 𝑓 = 𝑃 𝑖 , momentum is conserved. This is always true as long as there are no external forces.
While energy is also always conserved, this is not necessarily true of kinetic energy. We find two types of collisions:
Elastic collisions = kinetic energy is conserved
Inelastic collisions = kinetic energy is not conserved.
The greatest loss in kinetic energy occurs when two objects collide and then stick together. This is a “completely inelastic collision”.

11. Inelastic Collisions
For inelastic collissions, we cannot use energy conservation, so we used momentum conservation, 𝑃 𝑓 = 𝑃 𝑖 .
However, this doesn’t always lead to a unique answer.
For completely inelastic collisions, momentum conservation does give a unique answer
𝑃 𝑖 = 𝑝 1𝑖 + 𝑝 2𝑖 = 𝑃 𝑓 =𝑀 𝑣 𝑐𝑚
Thus, we can find the velocity of the objects once they have stuck together (and this works for any number of initial objects).

12. Inelastic Collisions
Quiz – a Prius (mass = 1300 kg) and a Chevy Suburban (mass = kg) are traveling towards each other at 120 km/hr. They collide and stick together. What is the speed of the resulting wreck?
20 km/hr
40 km/hr
60 km/hr
80 km/hr
About 2×106 J of energy is lost in the collision. Where does it go?
Do inelastic collision on air table. →

13. Inelastic Collisions
Quiz – suppose rain fall vertically into an open cart rolling along a straight horizontal track with negligible friction. As a result of the accumulating water, the speed of the cart
increases
does not change
decreases