1. Momentum
LT 4.1

2. What does momentum mean to you?

3. What is Momentum Really?
It’s when something is moving.
The more momentum an object has, the harder it is to stop.
What properties can make an object hard to stop? (Discuss)

4. What properties of an object can make it hard to stop?
Mass
Velocity
More mass = more inertia.
More velocity means it takes more [-] acceleration to get to stop.

5. Equation for Momentum
Based on mass….
And velocity….
𝑝 =𝑚 𝑣

6. An object’s velocity is tripled. How does the momentum change?
Divided by 3
No change
No way to tell

7. Same Mass / Different Velocity𝒗
𝒑 =
𝒑 = 𝒗 m m
Child Walking at 𝟏 𝒎 𝒔
Child Running at 𝟑 𝒎 𝒔

8. You fill a car with people until its mass is doubled
You fill a car with people until its mass is doubled. How does the momentum change when maintaining 20 mi/hr?
Doubled (x2)
Quadrupled (x4)
No change
No way to tell

9. Different Mass / Same Velocity
𝒑 =
𝒑 = 𝒗 𝒗 m
Full car at 𝟏 𝒎 𝒔
Empty Car at 𝟏 𝒎 𝒔

10. Momentum doubled from A to B
Compare momentum of each scenario: A: Empty wagon pulled very quickly. B: Full wagon with double mass pulled with half speed.
Momentum doubled from A to B
Momentum halved from A to B
No change
No way to tell

11. Different Mass / Different Velocity
𝒑 = m
𝒑 = 𝒗 m 𝒗
Full wagon pulled at 𝟏 𝒎 𝒔
Empty wagon pulled at 𝟐 𝒎 𝒔

12. Momentum changes. What could have caused that?
𝑝 =𝑚 𝑣
Change in mass
Change in velocity
Change in both mass and velocity
All of the above

13. Quick write: What actions could cause a change in momentum?
(2 min) Think of as many actions as possible that could cause a change in momentum.

14. Force Changes Momentum
If mass is constant, velocity must have changed.
We need to apply an impact force in order to change the velocity.
THEREFORE: A way to change the momentum is to change the impact force applied.

15. BUT
Impact force needs to be applied over a change in time to change the velocity (and therefore momentum)
This looks like:
𝐹 ∆𝑡= ∆ 𝑝

16. ∆ 𝑝 ∆ 𝑝 is referred to as “impulse” So, 𝐹 ∆𝑡= ∆ 𝑝 becomes:
𝐹 ∆𝑡=𝒊𝒎𝒑𝒖𝒍𝒔𝒆
Which can be:
𝐹 ∆𝑡= ∆ 𝑚 𝑣

17. Momentum 𝐹 ∆𝑡= ∆ 𝑚 𝑣 SUM-UP:
An impact force applied over a change in time will lead to a change in velocity
The change in velocity leads to a change in….
Momentum

18. Throw as FAST as you can.
Analyze what we’re doing to throw super fast.

19. Force is Constant Short Time for Impact Long Time for Impact
Throw from here.
Less time in contact means acceleration from force doesn’t last…. less ∆ 𝑣
Throw from here.
More time in contact means acceleration from force lasts longer…. More ∆ 𝑣

20. Practice

21. Conservation of Momentum 4.5

22. I. Inelastic Collision
Inelastic – not elastic. Crash and stick. Example: a football player tackles and holds on to another football player. Non-example: a bouncy ball bounces against a wall.

23. 1st Listen, Predict, Observe
Two cars (blue and red) have equal mass.
Blue car moves toward a stationary red car with 𝑣 , and they will stick together with Velcro.
How will the velocity of both cars stuck together relate to the blue car’s 𝑣 𝑜 ?
𝑣 𝑜 >0 𝑚 𝑠
𝑣 𝑜 =0 𝑚 𝑠
𝑣 𝑓 = ?

24. x 𝟏 𝟐
𝑣 𝑜 >0 𝑚 𝑠
𝑣 𝑜 =0 𝑚 𝑠
1 m s
4 m s
2 m s =2 =8 =4
𝑣 𝑓 = ?

25. Before collision: 1mass moving 8 𝑚 𝑠 After collision: 2mass moving 4 𝑚 𝑠
How did Momentum change from before to after the collision?
It doubled because mass doubled.
It was halved because velocity was halved.
It didn’t change.

26. Conservation of Momentum (CoM)
CoM – the idea that total momentum doesn’t change between colliding objects from just before to just after the collision.
𝑝 𝑇𝑜 = 𝑝 𝑇𝑓

27. Understanding how ratio between masses affects combined velocity.
Algebraic analysis

28. 𝑝 𝑇𝑜 = 𝑝 Tf K: 𝑚 𝐵 = 𝑚 𝑅 𝑣 𝐵𝑜 =8 𝑚 𝑠 𝑝 𝑅𝑜 =0 𝑚 𝑠 =𝑚 S:
𝑝 𝐵𝑜 = 𝑝 𝐵𝑓 + 𝑝 𝑅𝑓
𝑚 𝐵 𝑣 𝐵𝑜 = 𝑚 𝐵 𝑣 𝐵𝑓 + 𝑚 𝑅 𝑣 𝑅𝑓
Substitute:
𝑚 𝑣 𝐵𝑜 =𝑚 𝑣 𝐵𝑓 +𝑚 𝑣 𝑅𝑓 E:
𝑝 =𝑚 𝑣
𝑝 𝑇𝑜 = 𝑝 Tf

29. Solve for final velocity: 1 2 𝑣 𝐵𝑜 = 𝑣 𝑓 = 𝑣 𝑓K:
𝑣 𝐵𝑓 = 𝑣 𝑅𝑓
because the carts are together at the end of the collision… they have the same velocity!
𝑚 𝑣 𝐵𝑜 =𝑚 𝑣 𝐵𝑓 +𝑚 𝑣 𝑅𝑓
Substitute:
𝑚 𝑣 𝐵𝑜 = 𝑚 𝑣 𝑓 +𝑚 𝑣 𝑓
Combine:
𝑚 𝑣 𝐵𝑜 = 2𝑚 𝑣 𝑓
Solve for final velocity:
1 2 𝑣 𝐵𝑜 = 𝑣 𝑓
= 𝑣 𝑓
=4 𝑚 𝑠
(2 min) Describe our process to a partner.
(1 min) Describe how the ratio of starting to final masses affected our answer.

30. 2nd Listen, Predict, Observe
Red has 2 times the mass of Blue.
Blue car moves toward a stationary red car with 𝑣 , and they will stick together with Velcro.
How will the velocity of both cars stuck together relate to the blue car’s 𝑣 𝑜 ?
𝑣 𝑜 >0 𝑚 𝑠
𝑣 𝑜 =0 𝑚 𝑠
𝑣 𝑓 = ?

31. To reference last example
K: 𝑚 𝐵 𝑚 𝑅 𝑣 𝐵𝑜 =9 𝑚 𝑠 𝑝 𝑅𝑜 =0 𝑚 𝑠
To reference
last example =𝑚
=2𝑚
𝑝 𝑇𝑜 = 𝑝 Tf S:
𝑝 𝐵𝑜 = 𝑝 𝐵𝑓 + 𝑝 𝑅𝑓
𝑚 𝐵 𝑣 𝐵𝑜 = 𝑚 𝐵 𝑣 𝐵𝑓 + 𝑚 𝑅 𝑣 𝑅𝑓
Substitute:
𝑚 𝑣 𝐵𝑜 =𝑚 𝑣 𝐵𝑓 +𝑚 𝑣 𝑅𝑓 E:
𝑝 =𝑚 𝑣
𝑝 𝑇𝑜 = 𝑝 Tf

32. Practice.

33. A System
System – a set of related parts that form a whole We determine our own system for all problems. Usually, a system will include ~2 objects that are the focus of a word problem.

34. Two carts (red and blue) collide on a track. What should our system be?
Both carts
Both carts and the track
Both carts, the track, the table
Both carts, track, table, Earth

35. Types of Systems
Open System exists when BOTH of the following are true:
Closed System exists when at least 1 of the following is true:
Other objects that are not part of the system interact with the system in a significant way.
Time of interactions are very long.
Other objects not part of the system do NOT interact with the system in a significant way.
The shorter the time frame we look at, the more likely our system is closed.
“Significant” Interaction: changes our data by more than 10%

36. Which is an example of a closed system?
A single car rolling forward to a stop.
A person tackles another person.
A ball is thrown from a tall building.
You begin running down a sidewalk.

37. Check in!
Discuss the recent content with your partners!

38. Ball of Clay against Wall. CoM?
Clay has momentum before it hits the wall.
Clay has no momentum after it hits the wall.
How was momentum conserved?
Write down your ideas.

39. Clay vs. Wall Explanation
In each scenario, clay and wall and earth combine and move together.
In real life, the earth has a lot of mass, so the velocity is minimal.

40. Check in!
Discuss the recent content with your partners!

41. CoM Mathematics

42. 2 different CoM Scenarios:
Inelastic collision
Explosion

43. CoM: Total Momentum in a System is Saved
Crash!
Just Before Collision
Just After Collision
𝑝 𝑐ℎ𝑖𝑙𝑑 =60 𝑘𝑔 𝑚 𝑠 𝑝 𝑎𝑑𝑢𝑙𝑡 =0 𝑘𝑔 𝑚 𝑠 𝑝 𝑡𝑜𝑡𝑎𝑙 = 𝑝 𝑐 + 𝑝 𝑎 𝑝 𝑡𝑜𝑡𝑎𝑙 =60 𝑘𝑔 𝑚 𝑠 +0 𝑘𝑔 𝑚 𝑠 𝑝 𝑡𝑜𝑡𝑎𝑙 =60 𝑘𝑔 𝑚 𝑠
𝑝 𝑡𝑜𝑡𝑎𝑙 =60 𝑘𝑔 𝑚 𝑠 We can use this information to find the child and adult’s 𝑣 .

44. 𝑚 𝑐 =20𝑘𝑔
𝑣 𝑐𝑜 =3 𝑚 𝑠
𝑚 𝑎 =40𝑘𝑔
𝑣 𝑎𝑜 =0 𝑚 𝑠
Predict the velocity of the child and adult as they move backwards together.
3 𝑚 𝑠
2 𝑚 𝑠
1 𝑚 𝑠
0 𝑚 𝑠
Ooof. What is our 𝑣 𝑓 ?

45. Find the final velocity of the adult and child after the child collides with the motionless adult.
Need to know: Mass of adult and child. Velocity of child and adult before crash. Find: Velocity of child and adult after crash.

46. 𝑣 𝑎 =0 𝑚 𝑠 𝑣 𝑐+𝑎 𝑣 𝑐 Crash! 𝐹 𝑆𝑎 𝐹 𝑆𝑐
𝑣 𝑎 =0 𝑚 𝑠
𝑣 𝑐+𝑎
𝑣 𝑐
Crash!
𝐹 𝑆𝑎
𝐹 𝑆𝑐
Outside unbalanced forces exist (such as friction and air resistance (add them to the FBDs on the left), but there is a short collision time.
𝐹 𝑎𝑐
𝐹 𝑐𝑎
𝐹 𝐺𝑐
𝐹 𝐺𝑎
𝑝 𝑡𝑜 = 𝑝 𝑡𝑓
𝑝 𝑐𝑜 + 𝑝 𝑎𝑜 = 𝑝 𝑐𝑓 + 𝑝 𝑎𝑓
𝑚 𝑐 𝑣 𝑐𝑜 + 𝑚 𝑎 𝑣 𝑎𝑜 = 𝑚 𝑐 𝑣 𝑐𝑓 + 𝑚 𝑎 𝑣 𝑎𝑓
𝑚 𝑐 𝑣 𝑐𝑜 + 𝑚 𝑎 𝑣 𝑎𝑜 = 𝑚 𝑐 𝑣 𝑓 + 𝑚 𝑎 𝑣 𝑓
𝑚 𝑐 𝑣 𝑐𝑜 + 𝑚 𝑎 𝑣 𝑎𝑜 = 𝑣 𝑓 𝑚 𝑐 + 𝑚 𝑎
𝑚 𝑐 𝑣 𝑐𝑜 + 𝑚 𝑎 𝑣 𝑎𝑜 𝑚 𝑐 + 𝑚 𝑎 = 𝑣 𝑓
𝑣 𝑐𝑜 =3 m s
𝑚 𝑐 =20𝑘𝑔
𝑣 𝑎𝑜 =0 𝑚 𝑠
𝑚 𝑎 =40𝑘𝑔
Both objects move together after collision, so they have the same final velocity. 𝑣 𝑐𝑓 = 𝑣 𝑎𝑓 = 𝑣 𝑓 .
Substitute that 𝒗 𝒇 in here.
We now have a common term of 𝑣 𝑓 . Distribute.
Lastly, solve for the final velocity by dividing.
𝑣 𝑓 =1 m s
𝑚 𝑐 𝑣 𝑐𝑜 + 𝑚 𝑎 𝑣 𝑎𝑜 𝑚 𝑐 + 𝑚 𝑎 = 𝑣 𝑓

47. Momentum Class Practice
Let’s build some academic momentum for the semeter!

48. Next we are working:
In our notebooks

49. Write down your prediction about how they will move
Lorenzo is playing football as the Quarterback. Lorenzo has a mass of 80 kg. He throws the 0.43 kg football from a standing position North with a velocity of 30 m/s. Find Lorenzo’s final velocity.
Write down your prediction
about how they will move
after the throw!

50. Next we are working:
Voting Cards!

51. What must Lorenzo do with the ball for him to move backward faster during the throw?
Give the ball a larger final velocity.
Apply more force to the ball.
Find a way for the ball to apply more force to him.
All of the above!

52. Next we are working:
In our notebooks

53. Write down your prediction about how they will move after
Gary is playing football with some Secret Service friends. Assume Gary’s mass is 80kg, and that he is running North down the field at 3 m/s. If Gary catches the 0.43kg football which was also moving North with a velocity of 26 m/s, what will Gary and the football’s final velocity be after the catch?
Write down your prediction
about how they will move after
the catch!

54. A bullet of mass kg is fired vertically right below a wooden block of mass 0.3 kg. The bullet makes contact with the block at 320 m/s upwards. Find the velocity of the bullet/block system just after collision.

55. No more gary shit, change up football stuff until after football throwing demonstration.
Split into deceleration change in momentum, acceleration change in momentum, how to slow shit down without breaking, how to speed things up without breaking them DONE

56. Stopping an object

57. Egg-toss.
Goal: Determine which pair has longest egg displacement during the egg toss.
DO:
Get a lab apron.
Go to park.

58. Egg-toss Analysis What was the best technique for throwing?
Why do you think that?
What technique did NOT work well?
Why?

59. From last class:
Recall: 𝐹 ∆𝑡= ∆ 𝑝 If mass is constant, then the only variable that changes momentum is velocity. 𝐹 ∆𝑡= ∆ 𝑚 𝑣

60. These variables were allowed to change.
What we kept constant with the football:
𝐹 ∆𝑡= ∆ 𝑚 𝑣
Applied as much force as possible.
Football did not change mass.
These variables were allowed to change.

61. RECAP: Force Constant Short Time for Impact Long Time for Impact
Throw from here.
Less time in contact means acceleration from force doesn’t last…. less ∆ 𝑣
Throw from here.
More time in contact means acceleration from force lasts longer…. More ∆ 𝑣

62. What we kept constant with the egg:
𝐹 ∆𝑡= ∆ 𝑚 𝑣
These variables were allowed to change.
Egg did not change mass.
Throwing it kept the ∆ 𝑣 constant.

63. What we kept constant with the egg:
𝐹 ∆𝑡=𝒊𝒎𝒑𝒖𝒍𝒔𝒆
These variables were allowed to change.
Change in momentum.

64. ∆ 𝑣 and mass Constant impulse
Short Time for Impact
Long Time for Impact
Less time to stop the egg means there must be greater acceleration, and thus, more force applied.
More time to stop the egg means acceleration does not need to be large, and thus, there can be less force applied. 𝑭 ∆𝒕 = =
𝒊𝒎𝒑
𝒊𝒎𝒑 𝑭 ∆𝒕

65. Voting Card Questions for: Constant Impulse

66. Which car is safer for a driver during a head-on collision?
Sedan
Smart Car
Same safety
No way to tell

67. Which car do you expect will experience less force during collision?
Sedan
Smart Car
Same force
No way to tell

68. Which car experiences a smaller acceleration?
link
Sedan
Smart Car
Same force
No way to tell

69. Which car takes longer to stop?
Sedan
Smart Car
Same force
No way to tell

70. Sum up:
To stop the sedan as quickly as the smart car, more force must be applied.
Sedan has more inertia

71. Sedan Smart Car Same force No way to tell
Remember what we’ve learned about impulse, time, and force. If all safety features are the same, which car is more dangerous for a driver?
Sedan
Smart Car
Same force
No way to tell
= more 𝑎 experienced

72. Why? The smart car stops more quickly. 𝑹𝒆𝒄𝒂𝒍𝒍: ∆ 𝒗 ~ 𝒂
Smart car experienced greater ∆ 𝑣
𝑹𝒆𝒄𝒂𝒍𝒍: ∆ 𝒗 ~ 𝒂

73. Which car is safer for a driver?
Sedan
Smart Car
Same safety
No way to tell
5. Depends on the mass and the safety features of the specific car.

74. Wrap-Up:
3 – facts you learned 2 – interesting things 1 – question still