Change of Momentum

Change of Momentum Formula

To understand what a change of momentum is, we must first define momentum. Remember that momentum is a quantity given to an object due to its velocity \(\vec{v}\) and mass \(m\), and a lowercase \(\vec p\) represents it:

$$\vec p = m \vec v\mathrm{.}$$

The greater the momentum, the harder it is for an object to change its state of motion from moving to stationary. A moving object with significant momentum struggles to stop and on the flip side, a moving object with little momentum is easy to stop.

Therefore, assuming the mass of an object doesn't change, the impulse is equal to the mass times the change in velocity. Defining our final momentum,

$$\vec p_\text{f}=m\vec v_\text{f}\mathrm{,}$$

and our initial momentum,

$$\vec p_\text{i}=m\vec v_\text{i}\mathrm{,}$$

allows us to write an equation for the total change in momentum of a system, written as:

$$\vec{J}=\Delta \vec p = \vec p_\text{f}- \vec p_\text{i}=m(\vec v_\text{f}- \vec v_\text{i})=m\Delta \vec v,$$

where \(\Delta \vec p\) is our change in momentum, \(m\) is our mass, \(\vec v\) is our velocity, \(\text{i}\) stands for initial, \(\text{f}\) stands for final, and \(\Delta \vec v\) is our change in velocity.

Rate of Change of Momentum

Now, let's prove how the rate of change of momentum is equivalent to the net force acting on the object or system.

We've all heard that Newton's second law is \(F = ma\); however, when Newton was first writing the law, he had in mind the idea of linear momentum. Therefore, let's see if we can write Newton's second law a little differently. Starting with

$$\vec F_\text{net}= m \vec a$$

allows us to see a correlation between Newton's second law and linear momentum. Recall that acceleration is the derivative of velocity. Therefore, we can write our new force formula as

$$\vec F_\text{net}= m \frac{\mathrm{d}\vec v}{\mathrm{d}t}\\\mathrm{.}$$

It is essential to note the change that was made. Acceleration is just the rate of change in velocity, so to replace it with \(\frac{\mathrm{d} \vec v}{\mathrm{d} t}\) is valid. As the mass \(m\) stays constant, we see that the net force is equal to the rate of change of momentum:

$$\vec F_\text{net} = \frac{\,\mathrm{d}(m\vec v)}{\mathrm{d}t} = \frac{\mathrm{d} \vec p}{\mathrm{d} t} .$$

We can rearrange this to get

\[\mathrm{d}\vec{p}=\vec{F}_\text{net}\,\mathrm{d}t.\]

With this new outlook on Newton's second law, we see that the change of momentum, or impulse, can be written as follows:

\[\vec{J}=\Delta\vec{p}=\int\,\mathrm{d}\vec{p}=\int\vec{F}_\text{net}\,\mathrm{d}t.\]

• The change of momentum, or impulse (represented by the capital letter \(\vec J)\), is the difference between a system's initial and final momentum. Therefore, it is equal to the mass times the change in velocity.
• Newton's second law is a direct result of the impulse-momentum theorem when mass is constant! The impulse-momentum theorem relates the change of momentum to the net force exerted:

  $$\vec F_\text{net} = \frac{\mathrm{d} \vec p}{\mathrm{d} t} = m\frac{\mathrm{d}\vec v}{\mathrm{d} t} = m\vec a.$$

• As a result, the impulse is given by\[\vec{J}=\int\vec{F}_\text{net}\,\mathrm{d}t.\]

In physics, we often deal with collisions: this doesn't necessarily have to be something as big as a car crash – it can be something as simple as a leaf brushing past your shoulder.

The momentum of a collision system is always conserved. Mechanical energy, however, does not necessarily have to be conserved. There are two types of collisions: elastic and inelastic.

Elastic Collisions and Momentum

First, we'll talk about elastic collisions. "Elastic" in physics means that the system's energy and momentum are conserved.

This entails that the total energy and momentum will be the same before and after the collision.

Fig. 4 is a great example of an elastic collision in action. Notice how the motion transfers completely from the left object to the right one. This is a fantastic sign of an elastic collision.

Inelastic Collisions and Momentum

Now to the far-from-perfect evil twin.

An example is throwing a piece of gum into a trash can floating in space (we specify that it is in space because we do not want to deal with the rotation of the Earth in our calculations). Once the gum takes flight, it has a mass and a velocity; therefore, we are safe to say that it also has momentum. Eventually, it will hit the surface of the can and will stick. Thus, energy is not conserved because some of the kinetic energy of the gum will dissipate to friction when the gum sticks to the can. However, the system's total momentum is conserved because no other outside forces had the chance to act on our gum-trash can system. This means that the trashcan will gain a bit of speed when the gum collides with it.

The Variable Change of Momentum of a System

All of the examples of collisions above involve constant impulse. In all collisions, the system's total momentum is conserved. A system's momentum is not conserved, however, when that system interacts with outside forces: this is a critical concept to understand. Interactions within a system conserve momentum, but when a system interacts with its environment, the system's total momentum is not necessarily conserved. This is because in this case, there can be a non-zero net force acting on the system, giving the whole system a non-zero impulse over time (through that integral equation we wrote down earlier).

Examples of Change in Momentum

Now that we know what the change of momentum and collisions are, we can start applying them to real-world scenarios. This wouldn't be a collision lesson without car crashes, right? Let's talk about how the change of momentum plays a role in collisions – first, an example.