1. Galileo found that a ball rolling down one incline will pick up enough speed to roll up another. How high will it roll compared with its initial height if (a) there is no friction and (b) if there is? 2. Many automobile passengers suffer neck injuries when struck by cars from behind. How does Newton’s law of inertia apply here? How do headrests help to guard against this type of injury?

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1. Galileo found that a ball rolling down one incline will pick up enough speed to roll up another. How high will it roll compared with its initial height if (a) there is no friction and (b) if there is? 2. Many automobile passengers suffer neck injuries when struck by cars from behind. How does Newton’s law of inertia apply here? How do headrests help to guard against this type of injury?

Mathematics
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I know that for (1) without friction the ball will always reach the same height every time.
You are correct. Have you studied potential and kinetic energy?

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Yes now we are learning about forces.
Like Newton's 1st law of motion.
Good. For number 1 justify for your answer using potential and kinetic energy. Remember that energy is conserved -- it can not be created nor destroyed. It is just transformed: |dw:1436917896804:dw|
For # 2, an object in motion will stay in motion unless altered by an external force An object at rest will remain a rest unless altered by an external force. In an accident, the head is initially at rest and wants to remain at rest. What is the headrest doing, though, during the accident? Is it in motion?
I feel like it is moving forward with the max force?
The headrest is moving with the car. The head is not. To prevent the neck from injury, the head needs to be prevented from being "pulled" back by it's tendency to remain at rest. The headrest serves this function. Make sense?
So the headrest forces the head to move forward ?
Not so much as preventing the head from remaining at rest to an extreme that it pulls on the neck. We are saying about the same thing. The difference is that I am applying Newton's law to explain what is happening to the head and how the headrest comes in to prevent the head from remaining too much "at rest." If the head were to move with the car during an accident, then no headrest would be necessary. Do you see that?
What do you mean by "pulls on the neck"?
Think of the head as a rock and the neck a toothpick. If you move the body suddenly forward, you can break the toothpick, because the rock wants to stay at rest. The neck is a very weak link to the body. Does this help?
@calculusxy if you get jerked forward, things tend to want to keep moving with a constant linear velocity, so your head get swung back -- the neck rest is there to prevent you from hurting your neck and damaging your central nervous system
okay makes sense
What about for the first question, where there will be friction?
your head is coasting alone with your car at a constant velocity, then your car gets jerked forward by getting rearended -- inertia makes your head resist this change in motion so you need something (preferably soft) to keep it moving with the car, much like the rest of the seat does to your body
this is why you feel yourself get pressed into your seat when you accelerate, too -- inertia
For #1 if there is friction, total energy doesn't change - you still have potential energy being converted into kinetic energy, but friction is heat, the surface of the ball and of the slide gets hot, so some energy will go to these two surfaces. There is also air drag, but that is negligible. There is also energy lost to sound, also negligible. Since you have all these losses in energy, the original potential energy it had at the beginning is lost and the ball can not reach its max potential, so to speak. Make sense?
That was for 1 b)

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