**Note 1**: This section deals with **linear momentum**. *Angular* momentum (the momentum of turning or spinning objects) will be covered in another section.

**Note 2**: In discussions about momentum and collisions, the concept of **kinetic energy** often comes up, as it will on this page. You can still learn a lot about momentum without knowing anything about **KE**, but once you do learn that, you might want to revisit.

**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

pianomoving at 10 mi./hour or by afeathermoving at the same speed?

Momentum couples velocity to mass to give us a better gauge 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 (Kg) × velocity (m/s) = **Kg·m·s ^{-1}**.

**Kinetic energy**, **KE = ½ mv ^{2}**, is also a function of mass and velocity, but it is a

X
### SI Units

SI stands for Le **S**ystème **I**nternational 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** (**p**) is mass x velocity: **p = mv**

The SI **units** of momentum are **Kg·m·s ^{-1}**.

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.

The only things that matter about any vector are its length and direction. The length, or **magnitude**, of a momentum vector is *how much* momentum there is, and its direction is the direction of the momentum, or its velocity component.

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.

You can think of a momentum vector as a velocity vector multiplied by a scalar mass.

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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.

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

For example, consider the drawing of billiard balls below. The system would be the balls and the table. The white ball is hit with momentum **P** into the stationary red balls, all packed together, touching, and initially at rest. 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, which *is* moving.

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, **p _{1}, p_{2}, ..., p_{11}**, is equal to the length of the initial momentum vector

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

X
### 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 **p _{1}** as it simultaneously strikes the two green balls at rest (

We denote the length of vector p_{1} with absolute-value bars, | |. In the context of vectors, these always mean "length of."

Conservation of momentum says that **|p _{1}|** before the collision must equal

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,

_{1}| = |p_{2}| + |p_{3}|

There are two basic types of collisions between objects, **elastic collisions** (also known as ideal collisions) and **inelastic 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: **m _{1}v_{1} = m_{2}v_{2}**.

In all collisions, momentum is conserved. That is, the total amount of momentum present in the system (**m _{1}v_{1} + m_{2}v_{2}** 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.

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 to the surroundings.

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.

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**, **v _{1}** &

The incoming velocity of the ball is **v _{1}**, and the outgoing velocity is

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 **v _{2}^{2}** 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} V^{2}**, 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 **v _{1} = - v_{2}**.

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.

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, **p _{pink} = -p_{green}**

*Image: Wikipedia Commons*

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.

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

_{engine} = mv_{1} + mv_{2} + mv_{3} + ...

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

_{rocket} = - p_{engine}

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.

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, p_{1} = m_{1}v_{1} and p_{2} = m_{2}v_{2}. 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 m_{1} + m_{2}, and a new velocity. That velocity has to be smaller than v_{1} because the total momentum is conserved but the mass of the moving object has increased.

Let's pause here to derive a new way to define kinetic energy, in terms of momentum. We already have **KE = ½mv ^{2}**. We'll begin with the definition of momentum:

Now square both sides to get

The right side is looking like ½mv^{2}, 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:

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

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 2^{nd} car be after the collision?

**Solution**

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 P_{tot} = -1300 Kg·m/s. We are given the momentum of the pink ball (cart) after the collision, **P _{1} = (130 Kg)(-13.5 m/s) = -1755 Kg·m/s**:

These two momentum vectors must sum to the total system momentum. Plugging in P_{1} = -1755 Kg·m/s and rearranging, we can find P_{2}:

Now P_{2} = m_{2}v_{2}, 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.

A 100 Kg ball traveling at 10 m/s to the right collides head-on (i.e. in one dimension) with a 120 Kg ball traveling to the left at 5 m/s, as shown in the figure below. Calculate the post-collision velocities of the objects.

**Solution**

Momentum

The momenta of each ball are calculated like this:

So the total momentum of the system before the collision (a vector sum) is

That momentum must remain constant in this elastic collision.

Energy

The kinetic energy of the first ball is

and the second ball:

The total KE of the system is a sum of positive scalars:

Now after the collision, it must be true that the momenta will sum to 400 Kg·m/s. I'll write the mass values in from here on:

And the total kinetic energy of this elastic collision must also remain the same, so we have:

This we can reduce a bit to

Now we have two equations, (1) and (2), and two unknowns, the velocities v_{1} and v_{2}. We can solve for v_{2} in (1) and simplify to get

Now we can plug that value of v_{2} into (2):

To simplify and solve for v_{1}, let's first square that denominator and move it outside the parentheses:

That fraction reduced easily:

Now let's multiply through by 3/5 because the coefficients are divisible by 5 and it will clear the 5/3:

Multiplication gives:

Now expand the binomial (20-5v1)2 :

Gathering terms gives a solvable quadratic:

We can reduce again by dividing by 5:

Now let's just complete the square to solve this exactly. Dividing by 11 gives:

Adding the square of 1/2 the coefficient of v_{1} to both sides, we get

Identifying the perfect square on the left and getting a common denominator on the right gives

Almost there,

Now we can take the square root of both sides and move the 20/11 to the right to isolate v_{1} to find two possible solutions.

We'll rule out the first solution because it's physically impossible (see the diagram above!). So the final velocity of the pink ball (100 Kg) is -6.26 m/s. Plugging that into our earlier expression for v_{2} gives:

Now let's use those velocities to check whether we do indeed end up with the same momentum as before the collision:

... and we do. The final picture looks like this:

Whew! That's a lot of work, but very satisfying!

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.

**Solution***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 **p _{1x} + p_{2x} = -0.744 - 0.39 = -1.134 Kg·m/s**. The total momentum in the y-direction after the collision is

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

Consider a low-speed coupling of two train cars. The car on the left collides with, then couples to the stationary car on the right. Caclulate the velocity of the coupled pair of cars after the collision, assuming no loss of velocity due to friction.

**Solution**

_{1} = m_{1}v_{1} = (30,000Kg)(8 m/s)

_{1} = 240,000 Kg·m/s

Now that's the momentum of the system after the collision, too.

The total mass of both cars together is **m _{1} + m_{2} = 60,000 Kg**, so the velocity of that double-car is calculated by rearranging the momentum formula,

and plugging in the numbers:

If you think about it, it makes sense that the velocity is halved in this case.