Linear momentum
Newton’s laws, we mentioned that the second law is given by
The net force acting on an object is equal to the time rate of change of its momentum.
Linear momentum
Linear momentum is given by: \[\begin{equation} \vec{p} := m \vec{v}. \label{eq:momentumdefined} \end{equation}\] where \(m\) is mass, a scalar, and \(\vec{v}\) is the velocity. Like energy (e.g. \(KE=\frac{1}{2}mv^2\)), momentum involves both mass \(m\) and velocity \(v\). Unlike scalar energy, momentum is a vector quantity and has the same direction as the velocity. It is measured in kg m s−1.
When we discussed Newton’s laws, we used \[eq:momentumdefined\] to obtain a simplified form of the second law for an object of constant mass: \[\begin{align} \sum \vec{F} &= \dfrac{d \vec{p}}{d t} \\ &= \dfrac{d (m\vec{v})}{d t} \\ &= \cancelto{0}{\dfrac{dm}{dt}} v + m \dfrac{d \vec{v}}{d t}\ \text{by chain rule} \\ &= m \dfrac{d\vec{v}}{dt} \\ \sum \vec{F} &= m \vec{a} \end{align}\]
In this unit we will look a little more at this quantity (linear) momentum.
Motivation
In kinematics, the three vector quantities position, \(\vec{x}(t)\), velocity \(\vec{v}(t)\), and acceleration \(\vec{a}(t)\) describe the movement of an object, but don’t tell us much about things like the object’s size or shape. The force \(\vec{F}\) acting on an object might be imposed on it from something, or it might be the result of some interaction of the object and the surroundings; in the most general case the force could be totally independent of things like the object’s mass.
We might like to have some quantities that could help us understand the “impact” an object’s motion might have on stuff it hits; a quantity that could help us get a feel for how hard the object was to get moving, or how hard it would be to make it stop. Enter momentum.
Tackle this: a thought experiment
By whom would you rather be tackled?
A linebacker (or defenseman, Great Dane) at full gallop
A linebacker (or defenseman, Great Dane) at a leisurely stroll
Carly Rae Jepsen (or kicker, Chihuahua) at full gallop
Carly Rae Jepsen (or kicker, Chihuahua) at a leisurely stroll
Most of us would answer D, but why? There are two main reasons; first, Ms. Jepsen’s (or a football kicker’s, or a Chihuahua’s) mass is much less than that of a linebacker; and second, at a leisurely stroll rather than a full gallop (i.e. at lower velocity) it will hurt less. Thus, if we want to get quantitative, our answer should hinge on something combining mass and velocity.
We could multiply the two, and it gives us momentum, i.e.: \[\begin{align} \text{momentum} &= \text{mass} * \text{velocity}, \\ \vec{p} &= m \vec{v}. \end{align}\] The resulting quantity momentum (\(\vec{p}\)) is a vector that points in the same direction as the velocity \(\vec{v}\). Its units are kg m s−1, or equivalently, N s. As mass \(m\) gets bigger, momentum increases. Similarly as \(|\vec{v}|\) increases, momentum also grows bigger. Aside from it being a vector, which means it is several numbers and not just one, it would appear to be useful in deciding by whom we wish to be tackled.
How to compute momentum
Xuan the Guide Dog Puppy (mass 65, 29.4 kg) is in the park when he is given the command come! He begins running at 8 m s−1 towards his handler. What is Xuan’s momentum?
Solution
It is straightforward substitution to compute Xuan’s momentum; but for exam purposes you guys will want to write out all the steps and show exactly what you are thinking!
First, identify that this is a momentum problem and so the governing equation you need to use, straight from your notes sheet, is \[\begin{equation} \vec{p} = m \vec{v}\ \text{(definition of momentum)}. \end{equation}\]
Next, you will identify from the problem statement the various known quantities, like \(m=\qty{29.4}{\kilo\gram}\) and \(\vec{v} = \qty{8}{\meter\per\second}\ \text{towards the handler}\). Substitution gives \[\begin{align} \vec{p} &= m \vec{v}\\ &= (\qty{29.4}{\kilo\gram})*(\qty{8}{\meter\per\second}\ \text{towards the handler}) \\ &= \qty{235}{\kilo\gram\meter\per\second}\ \text{towards the handler} \end{align}\]
What does it mean? How can I use momentum?
When introduced to a new physical quantity, it’s a good idea to ask yourself what it means and how it can be used. Momentum is the product of mass and velocity. A bigger, faster object has more momentum than a small, slow one. Momentum as a vector also encodes something about the direction the object is going. Compared to a small object, a massive object takes more effort to get moving, when it is moving it has more momentum, and it takes more effort to get it to stop. In situations with no outside forces acting, momentum is conserved (more on that below). Momentum helps us understand how a moving object might “impact” things that it hits.
Elastic and inelastic collisions and explosions
Momentum can be used to analyze collisions and explosions. A collision is a model for an interaction where the forces exerted between the involved objects in the system are much larger than the net external force exerted on those objects during the interaction. As only the initial and final state of a collision are analyzed, the object model may be used to analyze collisions. An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.
We will do more on this later in physics, but momentum and momentum conservation are great for understanding collisions and explosions. The general idea is that the linear momentum of a system of particles is the same before and after they collide. This is useful in considering things like billiard balls (pool), bocce balls, bullets, bricks thrown out of go-karts, particles, etc. Why not just study the forces at the moment of impact? Such impacts are often short and hard to study.
Fluid mechanics, rocket science
Momentum shows up in control volume continuum approaches to fluids, where we are not tracking individual fluid particles but are instead tracking the momentum crossing into or out of the control volume, or momentum diffusing through a fluid due to the effects of viscosity. We probably won’t get to fluid mechanics in , but keep it in mind if you plan to go further in science or engineering.
In rocket propulsion, propulsive thrust is generated by expelling mass out the back of the rocket at high speed. For the same reasons momentum is used to understand collisions and explosions, it can provide understanding here by considering impulse \[\begin{equation} \Delta \vec{p} = \vec{F} \Delta t = m \Delta \vec{v}, \end{equation}\] which provides a handy connection between impulse, the amount of mass being thrown and its relative velocity. Momentum is also useful when integrating equations of motion, i.e. \(\sum\vec{F} = \dfrac{d\vec{p}}{dt}\) form of Newton’s second law.
Conservation of momentum
The idea that momentum remains constant in the absence of external forces is a useful one. In physics, when we can say something stays constant, it provides a useful clue in solving problems or understanding what is physically happening; we say such quantities are conserved or that they follow a conservation law.
Other quantities that are conserved include charge, mass, energy, angular momentum, and some others (when you get to super super advanced physics).
See also
- list here