Activity: Momentum

Aims

  • Define momentum.
  • Define impulse.
  • Understand the relationship between rate of change of momentum and force.
  • Understand the principle behind crumple zones and dynamic belaying.
  • State Newton's 3rd law of motion.
  • Apply Newton's 3rd law to different situations.

Momentum

mass x velocity

  • What are the SI units of momentum?
  • Is momentum vector or scalar?

Impulse

Change of momentum

Two balls collide as shown

  • Calculate the impulse for each ball. The mass of the red is 1 kg and the grey 4 kg

    -3.6 kgms-1 for the red and 3.6 kgms-1 for the grey.

  • What does your answer tell you about the change of the total momentum?

Newton's second law and momentum

Let's have another look at the constant acceleration example.

  • Write an equation for the acceleration in terms of v, u and t.
  • Apply Newton's second law to get an equation relating F and a.
  • Substitute for a from the first equation.

You should now see why:

The rate of change of momentum of a body is directly proportional to the unbalanced force acting on that body and takes place in the same direction.

So for the case above

F space equals space fraction numerator m v space minus space m u over denominator capital delta t end fraction

This is a more general statement of Newton's 2nd law that applies even if the mass is changing.

Water jets

I tried making a water pistol in Algodoo

  • Apply newton's first law to explain why the water experiences a force when it hits the wall.
  • If the velocity of the water is v before it hits the wall and 0 after what is the change of velocity of the water?
  • If the mass of water hitting the wall per second is m/t what is the change of momentum per unit time?
  • What is the force exerted by the wall onto the water?

Graphical treatment

If we plot a graph of force vs time we can see that the area under the graph = FΔt. But we know from Newton's 2nd that F = (mv-mu)/Δt so FΔt is the change in momentum or impulse.

The area under and F vs t graph is equal to impulse.

The falling climber

Rock climbers sometimes fall off like I do in this clip.

Notice how far I fall, this isn't because the person holding the rope was asleep but because the rope is elastic and the person holding the rope (actually he was called Per not person) moved forwards. The reason for this is to reduce the force needed to slow me down, its called dynamic belaying. You can do an experiment to see how this works.

Use a force sensor to measure the force required to slow down a small mass after it has fallen a distance of about 20 cm attached to a string and a rubber band.

Plot graphs of F vs time and find the area under the curve in each case.

This could also be done in Algodoo.

The same principle is used in the crumple zone of a car. Find out what that is.

Newtons 3rd law of motion

If body A exerts a force on body B then body B will exert and equal and opposite force on body A.

It is important to note that this is about 2 bodies.

  • Is it possible for a body to experience a force without the interaction of another body?

Some examples

Free fall

When body A falls it is attracted to the ground by the Earth, what does newt3 tell us about the Earth?

Box on the floor

Here are the forces on a box on the floor.

  • What does newt3 tell us about the forces involved?

You might have thought that newt3 tells us that N = G but this is wrong. That's newt1, if the body is at rest the forces are balanced. newt 3 says that if the ground is pushing up on the block (N) the block must pull down on the ground and if the earth is pulling the block down then the block must be pulling the earth up.

Here are the forces on the Earth

Water jet

Earlier in this activity we looked at a water pistol. You might have noticed that in that example the wall was moved backwards. Use newt3 to explain why the wall moved.

Conservation of momentum

Consider a collision between two balls

  • Deduce the change in momentum of each ball.
  • If the time colliding = Δt write an expression for the force experienced by each ball?
  • According to newt3 the force on A must be equal and opposite to the force on B, use this fact to show that:

 m subscript 1 u subscript 1 plus m subscript 2 u subscript 2 space equals m space subscript 1 v subscript 1 plus m subscript 2 v subscript 2

We can deduce from this equation that the total momentum before the collision = total momentum after. In other words momentum is conserved

For an isolated system of bodies the total momentum is always the same.

Conservation of momentum in Algodoo

In reality it is difficult to obtain an isolated system, balls on a pool table are not isolated since they interact with the table. In Algodoo you can switch off friction.

  • Place two equal size balls on the ground and reduce friction to zero in materials.
  • You can set the velocity of each ball by right clicking and dragging the velocities box to one side.
  • To display the momentum select “information” and drag to the side.

Try colliding different masses at different velocities and check that momentum is always conserved (add up the momentum before and after the collision and show that it is the same), it will be of course since the programme is made to do so.

  • Keeping masses and velocities the same try varying the restitution and attraction. Making attraction negative will cause the balls to fly apart.
  • Is momentum always conserved?

Solving problems

As you have seen there are many outcome for a collision between two balls, to find the momentum of one of the balls after a collision you need to know the momentum of the other (or some more information about the collision that we will come to later).

Find the unknown velocity in the collision below

Solve the problems on page 71

Momentum problems

Newton's 3rd law multiple choice

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