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Collisions and Vectors
and Newton's Third Law

Clickers Channel 1

Saturday, 19 March 2011


Calibrated Peer Review
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Please at least try writing something with equations and uploading it well before the deadline. In case there are some horrible last-minute computer problems.

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Please...

·

At the end of the lecture, remember to turn your clickers off!

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Another special case
· · ·
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Collisions are where things bump or (to a physicist) when things split up.The important thing is that the interaction between the objects is brief. Learning about them involves Newton's Third Law (the trickiest one). And it involves vectors.


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Crash question
· The magnitude of the force applied by the train to the car during the collision is...
1. much bigger than the magnitude of the force applied by the car to the train. 2. A little bigger. 3. The same 4. A little smaller 5. Much smaller 6. Not enough information provided

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They are identical
· The car changes its speed far more. · But the change in momentum is equal to change in speed times mass - and its mass is far less. · Multiply mass and speed change together and you'll get the same answer for both car and train.

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VPython simulation

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Collision Problems
· · · ·
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So what do you do in a collision situation? The first thing is use conservation of momentum. Remember - momentum (if your speed is v not close to the speed of light) is p = m In all collisions, total momentum is conserved.


Train hits car...
Mt V
t

Mc

Initial state - train moving, car at rest. Momentum is pi=MtVt

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Final state
Mt Mc V

Momentum is pf=(Mt+Mc)V Momentum is conserved, so pi = pf, and MtVt=(Mt+Mc)V which you can solve to get (for example) the final speed. Or if you know the final speed, you could use it to get the starting speed, or perhaps one of the masses.
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Write down momentum
· · · ·
Saturday, 19 March 2011

Write down the momentum before the collision. Write down the momentum after the collision. Set them equal to each other. Solve for whatever you want to find out.


Clicker Question
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A ice-skater moving at speed v collides with an identical stationary iceskater, and the two stick cling after the collision. What is their velocity after colliding? 1. 2. 3. 4. 5. 6. v v/2 0 -v/2 -v Need more information

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Answer
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v/2. Because total momentum is conserved, we have mv=2mu, which gives a final velocity u=0.5v

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What if things bounce?
· ·
We assumed that the car and train moved at the same speed after the collision - i.e. they stuck together. But what if things bounce?

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VPython simulation

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Momentum conservation
· · ·
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We can still use momentum conservation but it's not enough. We have two unknowns (the velocities of the two objects after the collision) and only one equation. We need another equation.


Inelastic Collision
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Look at the relative speed at which the objects approach each other before the collision and move apart after the collision. If the speed afterwards is zero, they have stuck together. The collision is said to be totally inelastic. Kinetic energy has been transformed into deforming and heating the objects.

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Elastic Collision
· · ·
If they bounce apart at the same relative speed as they approached, the collision is said to be elastic. In this case, the initial kinetic energy is briefly transformed into spring energy, then comes back as kinetic energy. So final kinetic energy equals initial - no energy is converted into other forms.

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· · · · ·

Coefficient of Restitution
In-between cases... Look at the relative speed at which the objects approach each other before the collision and move apart after the collision. c is called the coefficient of restitution and will be between 0 (totally inelastic) and 1 (elastic). Some energy is lost unless c=1. You will have a lab on this.

Saturday, 19 March 2011


Clicker question
·
A person attempts to knock down a large wooden bowling pin by throwing a ball at it. The person has two balls of equal size and mass, one made of rubber and the other of putty. The rubber ball bounces back, while the ball of putty sticks to the pin. Which ball is most likely to topple the bowling pin?

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Answer
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Because momentum is conserved in these interactions, more momentum is transferred to the bowling ball from the rubber ball than from the putty ball. Hence the rubber ball is more likely to knock the pin over.

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Clicker Question
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Think Fast! You've just driven around a curve in a narrow, one-way street at 40 km/ hr when you notice a car identical to yours coming straight towards you at 40 km/hr. You have only two options: hitting the other car head-on or swerving into a massive concrete wall, also head-on. In the split second before the impact, you decide to...

Saturday, 19 March 2011


Answer
·
It makes no difference.Your change in momentum is the same in both cases. Imagine holding a thin sheet of metal between you and the oncoming car. The sheet will stay put (just as the wall does) because your momentum and that of the other car add up to zero.

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What if the collision is in 2D or 3D? M
2

· · ·

?
M1

You need to use the fact that momentum is a VECTOR Add up the initial momentum vectors to get the final momentum vector Very common error is to treat momentum as a scalar in situations like this.

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Two ways of handling vectors
1. Graphically (good for making sure answers make sense, and planning) 2. Components (good for getting precise numerical answers).

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Graphically
· Slide arrows around! · To add vectors, slide them around until the tail of one is at the head of another (i.e. put them in a chain). Sum is the vector from the first tail to the last head. · To subtract vectors, either add minus the vector (minus a vector is just the vector reversed), or slide them head-to-head. Difference is from the tail of the first to the tail of the second.

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Vector addition

a
b

1 2 3 4

a+b =?
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5 6. None of these


Answer
1

a+b =?

b
3

2

a

4

5 6. None of these

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b

Subtraction
1 2 3 4

a
a-b =?
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5 6. None of these


Answer

a-b =?

1 a
3

b

2

4

5 6. None of these

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Reversed subtraction b
1

a
b-a=?
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2 3 4

5 6. None of these


a b-a=?

b

Answer: none...
1 2 3 4

5 6. None of these

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Working out the Momentum Change
· Momentum points in the same direction as the velocity vector. · So sketch the velocity vector before and after some event, and the difference (vector difference) tells you the change in momentum, which will be the impulse applied.

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For example
· Bounce a ball off a wall. · What impulse (force times time - change in momentum) did the wall apply to the ball?

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So what is the impulse applied to the ball?

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This is the velocity vector before the impact. The momentum vector must point in the same direction.

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This is the velocity vector after the impact.

This is the velocity vector before the impact. The momentum vector must point in the same direction.

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So subtract this vector

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From this vector

So subtract this vector

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To subtract vectors, put them head-tohead, then draw the line from one tail to the other.

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Slide this one across...

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This is the change in momentum, and hence the impulse.

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B Screen
7 6 8

1

2 3 4

A

5 9 for zero magnitude

In what direction is my change in momentum from A to B? The impulse that must have been applied to me... Door
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Door

Assume I travel at a constant speed.


B Screen
7 6 8

1

2 3 4

A

5 9 for zero magnitude

In what direction is my change in momentum from A to B? The impulse that must have been applied to me... Door
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Door

Assume I travel at a constant speed.


B Screen

A

Momentum vectors at A and B

Door
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Door


B Screen

A

Momentum vectors at A and B

Door
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Door


B A

My final momentum (B) is equal to my initial momentum (A) plus the unknown impulse applied to me (I). So A+I=B I=B-A
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B

I

A

Move head-to-head, take vector from base of B to base of A.

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