Momentum vs. velocity


Momentum is very easily defined, it's mass × velocity, but why do we need it?

Consider this: Would you rather be hit in the face by a piano moving at 10 mi./hour or by a feather moving at the same speed? Momentum couples velocity to mass to give us a better gage of how much energy a collision can deliver.

The momentum of an object traveling in a straight line (linear momentum) is given the symbol p, and the definition p = mv. The SI units of momentum are mass × velocity = Kg·m·s-1.

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SI Units

SI stands for Le Système International d'Unités (French), or International System of Units.

It is a standardized system of physical units based on the meter (m), kilogram (Kg), second (s), ampere (A), Kelvin (K), candela (cd), and mole (mol), along with a set of prefixes to indicate multiplication or division by a power of ten.

Momentum

Momentum (p) is mass x velocity:   p = mv

The SI units of momentum are Kg·m·s-1

Momentum is a vector

Momentum, p = mv is mass (a scalar) multiplied by the velocity, a vector, therefore it too is a vector. Remember that multiplying a vector by a scalar can change its length and units, but not its direction.

Vectors have both length and direction, and those are the only things about them that matter. The length of a momentum vector is how much momentum there is, and its direction is the direction of the momentum.

Remember that vectors, like the two-dimensional ones on the right, can be moved around at will without loss of meaning, and that we add them head-to-tail, as shown here.

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Scalar

A scalar is a number, which has no direction implied, like mass, temperature or speed. Velocity is a vector that has both speed and direction.


Momentum is always conserved

The total amount of momentum in a system is always conserved. That is, momentum is never lost or gained.

For example, consider the drawing of billiard balls below. The white ball is hit with momentum P into the stationary red balls, all packed together, touching. We'll assume that the masses of all balls are the same, just to make things easy. Because they are not moving, the total momentum of all of the red balls is zero, therefore the momentum of the entire system is just the momentum of the white ball.

Shortly after the collision of the white ball with the lead red ball, the picture might look something like this (below). The white ball has lost most of its momentum, and each red ball has picked up a part of it. Each has scattered in a different direction, but the sum of the lengths of all 11 momentum vectors, p1, p2, ..., p11, is equal to the length of the initial momentum vector P.

All of the momentum is still there after the collision, it's just been redistributed.

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Conservation

In science, to say that a quantity is conserved means that, in a closed system, or in the universe, the amount of that quantity never changes, though it might get spread around in different ways.


Here's another illustration of what conservation of momentum means in terms of vectors. Consider the pink ball with momentum p1 as it simultaneously strikes the two green balls.

We denote the length of vector p1 with absolute-value bars, | |. In the context of vectors, these always mean "length of."

Conservation of momentum says that |p1| must equal |p2| + |p3|. We can never have more momentum in the green balls after the collision.

We might end up with less total momentum in the green balls, but that's because the energy of the collision can be distributed elsewhere, like sound or heat energy. In our ideal system,

|p1| = |p2| + |p3|


Collisions

There are two basic types of collisions between objects, elastic collisions (also known as ideal collisions) and inelastic collisions.

Elastic collisions

In an elastic collision, like the one illustrated below, two objects (the pink and green balls) approach each other with a certain momentum. We'll assume, for simplicity, identical masses and identical velocities, except for direction, so the momenta are the same: m1v1 = m2v2.

In all collisions, momentum is conserved. That is, the total amount of momentum present in the system (m1v1 + m2v2 here) is still present after the collision, except that it might be distributed a little differently.

In this collision, the balls collide and instantaneously reverse direction to head the other way. In an elastic collision, there is no deformation of the objects, so there's no perturbation of the atoms and molecules within, so there's no heat radiated away.

Collisions of certain real objects, such as billiard balls, are very nearly elastic. The atoms and molecules of certain gases collide pretty much elastically, too, which is a big help in calculating the properties and behavior of them using the ideal gas law.


Inelastic collisions

In an inelastic collision, momentum is still conserved in just the way it was for an elastic collision, but kinetic energy is not. Consider the collision in the drawing:

The situation is the same, but now the balls may deform as they collide, which can, in turn heat them up through atomic and molecular motion. Sound or even heat and light might be given off, carrying energy away. This inelasticity is more common for a real collision. This energy is lost to the surroundings. While energy is always conserved in the universe, it is not conserved in this system; some is lost from the two balls.

Elastic & inelastic collisions

In an elastic collision, all kinetic energy remains with the colliding bodies.

In an inelastic collision, some kinetic energy is lost to the surroundings in other forms, such as heat and sound.

In a perfectly inelastic collision, two objects collide, stick together and move as one object thereafter.

Elastic collision with a massive object

Let's consider a ball rolling into an immovable object, like a wall. If we let the wall be very massive compared to the ball, then the collision won't cause it to move. The wall won't have any momentum, either before or after the collision.

We'll let capital letters M & V stand for the mass and velocity of the wall, and lower case, m, v1 & v2 stand for the mass and velocity of the ball.

The incoming velocity of the ball is v1, and the outgoing velocity is v2. Our job here is to find v2 in terms of v1.

We begin with the total momentum of the system, before and after. Remember that these must be equal. Before the collision, the wall is stationary, so it has no momentum, so all of the momentum of the system is on the left side of this equation. On the right is the momentum after the collision. We'll allow for movement of the wall, then look at that later:

If we divide both sides by m, we get

Now let's put both velocities together on the left:

Now we're going to want to compare this momentum-balance equation to the kinetic energy one, so let's square both sides and save this equation with a ( * ) for now:

Now let's consider the kinetic energy of the system. It's the same process. The KE before the collision (left side of this equation) just involves the ball; the wall isn't moving. On the right, after the collision, we allow for movement of the wall.

If we multiply through by 2 we get

and here again, we can divide by m:

Moving v22 to the left, we get:

Now let's multiply both sides of that equation by M/m so that we can line it up with equation ( * ) above.

Now we have two equations containing (M/m)2 V2, so we can hook those up using the transitive property:

If we multiply both sides by m/M, we get

Now let's do something interesting. Let's ask what happens as we make the mass of the wall, M, infinitely large. The limit ("lim") notation below is used for that. This statement says, "in the limit where M becomes infinitely large, the expression goes to zero."

That is, as M gets huge, m/M gets very small, and the term on the right side of our equation vanishes, so we have

That gives us

If we take the square root of both sides we have

Now we know that the ball isn't going through the wall, so v1 = - v2.

This means that in a collision of a moving object with an immovable object, all of the momentum remains in the moving object and its velocity is just reversed.

The scenario is just a bit more complicated if the ball hits the wall at a non 90˚ angle, of course, but not much more complicated. We'll tackle that later.

Of course, this is only true for an elastic collision, where all of the kinetic energy is conserved. In a real collision, we always lose some energy to the generation of sound or heat.

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Transitive property

If a = c and b = c,

then a = b.


Conservation of momentum

Example 1: Explosions

Consider the setup below. If we can contrive a way to place a small explosive between two balls of equal mass at the center of a track, with bumpers on the ends to ensure as close as possible to elastic collisions, the two balls should bounce off either end with the same momentum, meet back in the middle and stop there. You can click the forward button to see an animation in slow motion.

This experiment (and it can be done as an experiment on an air-track, a track with very little dynamic friction) is a very good one for investigating conservation of momentum.

Think about it for a minute. At the beginning, the momentum of the system is zero. The law of conservation of momentum tells us that the momentum of the system must remain zero. Therefore the velocity of each ball after the explosion between them must be the same, but in opposite directions (vector velocities add to zero). After the bounce, the two balls collide with equal momenta but opposite direction, so the momenta add to zero and everything stops.

At all times in the animation, ppink = -pgreen


Image: Wikipedia Commons

Fireworks

Fireworks are a great example of conservation of momentum. The symmetry of fireworks explosions shows that momentum is conserved.

To achieve a pattern like this one, the explosion of the colored fireworks charges is timed so that it occurs right at the top of the flight of the fireworks package, where the velocity of the shell is zero. At that point the momentum of the system is roughly zero, so the explosion has spherical symmetry, in which all of the 3-D momentum vectors must add to zero.

That means there must be as many colored streaks to the right as to the left, as many up as down, and so on.


Conservation of momentum

Example 2: How rockets move

In the early days of rocketry, many believed that a rocket couldn't move in space because there was nothing for the rocket exhaust to "push against" in the vacuum of space. But conservation of momentum won out, and it turns out that rockets do just fine with nothing to push against.

Consider the picture below. The top figure shows a stationary rocket. The velocity of the rocket is zero, so its momentum is zero. No hot gas molecules are being ejected from the nozzle at the back, so there is no momentum there.

Now let's ignite the fuel, which causes the ejection of hot gas molecules to the left at very high velocities.

While the momentum of each molecule or atom is very small, there are a very great number of them, adding up to the left-pushing momentum

pengine = mv1 + mv2 + mv3 + ...

Now conservation of momentum says that the total momentum of this system (rocket plus gases) must remain zero, so there must be an equal momentum of the rocket toward the right, with

procket = - pengine

Here's an interesting thing: when it comes to real rockets, notice that as the fuel is ejected from the engine, the mass of the rocket actually decreases, so its velocity must increase proportionally in order to maintain the momentum. So as the fuel tank empties, the forward speed of the rocket increases even more.



Example 3

Perfectly inelastic collisions: coupling

In a perfectly inelastic collision, two objects collide along a line and they stick together, effectively forming one object. The classic example of this is the coupling of two train cars, like this:

In the animation (play it a few times), you can see that each train car has its own momentum, p1 = m1v1 and p2 = m2v2. The initial velocity of the second car is zero, so its momentum is zero. That means that all of the momentum of the system is in the first car.

After the collision, when the cars couple, they are effectively a single car with mass m1 + m2, and a new velocity. That velocity has to be smaller than v1 because the total momentum is conserved but the mass of the moving object has increased.

Kinetic energy in terms of momentum

Let's pause here to derive a new way to define kinetic energy, in terms of momentum. We already have KE = ½mv2. We'll begin with the definition of momentum:

Now square both sides to get

The right side is looking like ½mv2, so let's divide out one of the masses:

And finally, if we divide both sides by 2, we have:

So we have a new formula for the kinetic energy, on that comes in handy from time to time:

Two formulas for the kinetic energy

The kinetic energy of a moving object can be calculated in two ways:


Example calculations


Example 1

Elastic collision

Bumper cars at amusement parks bounce of one another in approximately elastic collisions. Let's say that a 130 Kg bumper car traveling at 10 m/s hits a 200 Kg bumper car moving at 13 m/s in the opposite direction. If the first car bounces back at a rate of 13.5 m/s, what must the velocity of the 2nd car be after the collision? collision?

Solution: First we ought to draw a simple diagram to organize what we know. It doesn't have to be fancy; we can just reduce everything to balls:

The velocity vectors and masses are shown. The green ball has a negative velocity. It doesn't matter which direction we choose as negative, just that they're opposite. We can calculate momentum vectors:

Now the total momentum is the sum of these oppositely-signed vectors:

The law of conservation of momentum says that the momentum of this system must remain forever Ptot = -1300 Kg·m/s. We are given the momentum of the pink ball (cart) after the collision, P1 = (130 Kg)(-13.5 m/s) = -1755 Kg·m/s:

These two momentum vectors must sum to the total system momentum. Plugging in P1 = -1755 Kg·m/s and rearranging, we can find P2:

Now P2 = m2v2, so

from which we can solve for the velocity of the second cart.

The second cart rolls away much more slowly, but that's necessary for momentum to be conserved. In reality, the collisions aren't completely elastic, and the post-collision velocities would be lower.


Example 2

A two-dimensional collision

Consider the diagram below. Two balls of equal mass but different velocities collide at the angle shown. If the final velocity of the green ball is 1.5 m/s and it is deflected by 105˚ to the right of its path (up in the figure), calculate the direction and speed of the pink ball after the collision.

Here's a picture of the situation before the collision. Balls of the same mass are colliding, and the velocity and momentum vectors are two-dimensional, with x- and y-components.

The first thing to do is to calculate the x- and y-components of the velocity vector of the pink-ball. We'll call the up and right directions positive and the down and left directions negative. It's not crucial which is which, just that we remain consistent throughout our work:

The x-component of the momentum is:

and the y-component of the momentum is:

(Yup, easy because of the 1Kg mass). The green ball has no vertical velocity, and therefore no vertical momentum, so its x- and y-momenta are

and

Now we can calculate the total momentum in the x- and y-directions:

and

The next step is to look at what we're given about what happens after the collision, keeping in mind that the total momentum in the x- and y-directions must remain constant. Here's the picture:

First we calculate the x- and y-components of the green ball velocity after the collision:

... and convert those to momenta by multiplying by the mass of the green ball:

and

Finally, the x- and y-momenta of both balls must sum to the total momentum in each of those directions before the collision (conservation of momentum), so we have:

Rearrangement gives us the momentum of the pink ball in the x-direction:

Likewise we can rearrange

to get:

Now let's check everything for consistency. The total momentum in the x-direction after the collision is p1x + p2x = -0.744 - 0.39 = -1.134 Kg·m/s. The total momentum in the y-direction after the collision is p1y + p2y = 1.45 - 0.95 = 0.5 Kg·m/s. These are the same x- and y-momenta as before the collision.

Here's a picture of the whole collision, drawn roughly to scale:


Coming soon: More examples, and a practice problem set with solutions.


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