--- ## Momentum basics - Momentum is the product of an object's **mass and velocity**. It is denoted using the variable $\small p$ and it uses the unit: $\small Kg\cdot m/s$. This unit is also sometimes referred to as a *"Newton second"* denoted: $\small N\cdot s$. - Since the velocity of an object is a vector quantity that can be either **negative or positive** it should follow that ==momentum is also a **vector quantity.**== The vector representing an object's momentum will **always point** in the **same direction as that object's corresponding velocity vector**. - The momentum vector will also **always be greater in magnitude than the velocity** vector. This should make sense because momentum is the *product* of an object's velocity ***and* mass**. Since mass can never be negative, momentum **must be greater in magnitude than its associated object's velocity**. - Just like with velocity, the momentum of an object can be deconstructed into it's x and y components which can be helpful when performing calculations. $\newline$ $\text{Momentum equations: }\;\;\large\color{LightCyan}\boxed{ \vec{p}=m\cdot\vec{v}}\;\;\;\;\;\boxed{\Delta\vec{p}=\vec{p}_{\small f}-\vec{p}_{\small i}}$ $\newline$ $\text{Component momentum equations: }\;\color{LightCyan}\boxed{p_x=mv_x}\;\;\;\;\;\;\;\boxed{p_y=mv_y}$ $\newline$ #### *Impulse momentum theorem* - The impulse-momentum theorem states that an **impulse exerted on an object will always produce a directly proportional change in that object's velocity.** In other words, impulse can be thought of as a measure of the **change in momentum of an object**. *Using this theorem we can derive a simple equation that relates both impulse and momentum.* $\text{\normalsize Impulse momentum theorem: }\;\;\;\Huge\boxed{{\;\;\large\begin{aligned}\vec{J}&=\color{LightCyan}\Delta \vec{p}\\\vec{J}&=\color{LightCyan}\vec{p}_{\small f}-\vec{p}_{\small i}\\\vec{F}_{avg}\;\cdot\Delta t&=\color{LightCyan}\vec{p}_{\small f}-\vec{p}_{\small i} \end{aligned}\;\;}}$ $\newline$ >[!bug] >##### *Always remember that velocity is a vector quantity!* >- If a ball bounces off of a wall and rebounds off of it with the same velocity that it had when traveling towards the wall, the change in velocity should be equal to **double the original velocity**. > - This is because velocity **is a vector quantity** and therefore will switch signs when the ball is traveling in the opposite direction. >- ==**If an object changes direction after it collides with another object don't forget to include the negative sign in that object's initial or final velocity when performing calculations with the above equations**.== If you don't do this you will end up **subtracting** that object's initial velocity from its final velocity when in reality you should be **adding the two values together** *(to find the change in the object's momentum)*. > >$\Delta p=p_{\small f}-p_{\small i}=m\Big(v_{\small f}-(-v_{\small i})\Big)=\color{Bisque}m\cdot 2v_i$ >[!tip] >##### *Force and the duration of a collision* >- Using the impulse momentum theorem we can derive a new fundamental relationship that tells us that the **average force needed to stop a *colliding* object is inversely proportional to the duration of the collision**. > - Using this relationship we can also mathematically prove another very intuitive concept: If the duration of the collision can be **increased, the force of the impact will be decreased**. > >$\text{\small Relationship between average force, time and momentum: }\;\;\color{Bisque}\vec{F}_{avg}=-\frac{\vec{p}_{\small i}}{\Delta t}=\frac{-mv_i}{\Delta t}$ #### *Total momentum* - Its often useful to quantify the **sum of the momenta of all the objects contained within a system**. This quantity is called the **total momentum of a system** and it is denoted using a **capital $\small\vec{P}$**. - *NOTE: A "system" is just an arbitrary group of objects.* - The total momentum of a system is just the **vector sum** of the momenta of each object contained within that system: $\text{Total momentum: }\;\vec{P}=\vec{p}_{\small 1}+\vec{p}_{\small 2}+\vec{p}_{\small 3}\;...$ $\newline$ --- ## *Conservation of momentum* - Momentum **can not be created or destroyed,** it can only ever be **transferred from one object to another**. ==Therefore we say that **the TOTAL momentum of an isolated system is conserved**.== In other words, the total momentum of a system *(the sum of each object's individual momenta)* will never change. - An *isolated system* is a system that experiences no **external forces**. Every force exerted on an object within an isolated system ==**must come from another object within that system**== otherwise the system is not isolated. - Forces that act only between objects within the system are called **internal forces**. Forces that originate from objects outside of the system are known as **external forces**. Using the above definition we can see that the objects within an isolated system will only ever experience **internal forces**. - In physics we say that a quantity is **conserved if it is the same before and after an interaction.** - If the net force on a system is 0 that system is isolated and therefore will abide by the law of conservation of momentum. $\color{}\Huge\boxed{\color{LightCyan}\large\text{Conservation of momentum: }\;\vec{P}_{f}=\vec{P}_i\;\;\;\text{\color{grey}or}\;\;\;mv_f=mv_i}$ $\newline$ ##### *What if the mass of an object changes?* - If the **mass of an object** somehow *changes during an interaction* the property of *conservation of momentum* implies that the **velocity of that object must also change accordingly.** This happens because the momentum of an object cannot change without some external force acting on it. In other words, ==changing the mass of an object will **force the velocity of that object to also change in order to keep its momentum constant.**== - *If the mass of an object is increased its velocity will decrease and if its mass is decreased, its velocity will increase.* >[!tip] >##### *Conservation of momentum and components* >- Even when we split momentum up into its individual components, the property of conservation of momentum still applies. In other words, **the x component of the system's total momentum will never change *(assuming it's an isolated system)* and likewise, the y component of the system's total momentum will also never change.** ##### *Solving conservation of momentum problems* - You know you are dealing with a *conservation of momentum* problem if the question provides information about the mass and velocity of objects **before and after some event** *(usually a collision)*. In most questions there will be a single unknown variable that you will be tasked to find, like the *velocity* of object __ after a collision, or the mass of object __ and so on... ###### *The general process for solving these problems:* 1. The first step is to choose an isolated system to perform all of your calculations on. If this is not possible you should at least try to choose a system *that is isolated during some part* of the problem *(meaning it is isolated for a period of time)*. 2. Next, draw a visual overview of the system before and after the interaction. 3. Now you can write out the conservation of momentum equations in terms of its **x and y components**. Isolate the variable for which your are trying to solve and then carry out the calculation. - Doing these calculations requires you to **first find the total momentum equation of the system right before and after some specific event**. - By setting these equations equal to each other you can solve for unknown variables. 4. Finally, asses if the answer you got appears reasonable. If not you know that you have most likely made some calculation error. ###### *Unique situations:* - If you are solving for some **unknown variable in an *in-elastic collision (see [[Springs and elasticity]])* its important to remember that *both objects* will be moving in the same direction *with the same velocity* after the collision**. This means you can effectively consider them as **one, single object** and combine their masses to calculate the system's final momentum. $\text{In-elastic collision example: }\;\;\;\left[\;\;\begin{aligned}\vec{p}_{\small i}&=m_1v_1-m_2v_2\\\vec{p}_{\small f}&=(m_1+m_2)v_{final}\end{aligned}\;\;\right]$ $\newline$ >[!example] >##### *Example conservation of momentum problem* >- A small, 100 g cart is moving at on a frictionless track when it collides with a larger, 1.2 kg cart at rest. After the collision, the small cart recoils at 0.85 m/s. What is the speed of the large cart after the collision? >--- >- The first step in any conservation of momentum problem **is to make an equation for the initial *total momentum* of the system and an equation for the final *total momentum* of the system.** > - *The system in this equation will contain only the 2 carts because there is no friction or gravity or any other forces involved.* > >$\newline$ >$\text{Total momentum equations: }\;\left[\;\;\begin{aligned}\vec{p}_{\small i\;total}&= 0.1\cdot 1.2\\\\\vec{p}_{\small f\;total}&=1\cdot v-0.85\cdot 0.1\end{aligned}\;\;\right]$ >$\newline$ >- Now that you have the equation for the total momentum of the system before and after the event *(which in this case is a collision)* we can set these two equations equal to each other *(due to the conservation of momentum property)* and solve for the unknown. **The unknown in this problem is $\small v$, the velocity of the large cart after the collision.** > >$\newline$ >$\text{Conservation of momentum: }\;\vec{P}_{f}=\vec{P}_i\;\;\longrightarrow\;\;v-0.85\cdot0.1=0.1\cdot1.2$ >$\therefore\;v=\frac{0.1\cdot1.2+0.1\cdot 0.85}{1.2}=\color{Bisque}0.17\,\frac{m}{s}$ ##### *Force and momentum* - When the **momentum** of an object changes a **[[Forces|force]]** is exerted onto that object which causes it to **accelerate in a direction**. - The **amount of force** exerted on a system is equal in magnitude **to the change in the momentum of that system per unit of time**. The amount of time it takes for the momentum to change and the **resulting force generated are inversely related**. *(less time = more force, more time = less force)*. - We can calculate the force exerted on a system due to the change in momentum over time with the following equation where $\small m$ is the *mass* of an object, $\small v$ is it's velocity and $\small t$ is the time. $\text{Force due to momentum: }\;\;\Huge\boxed{\large\color{lightcyan}F=\frac{[mv]_{\small f}-[mv]_{\small i}}{t_{\small f}-t_{\small i}}}$ --- ## Angular momentum - Angular momentum is exactly like **linear momentum** except instead of quantifying the product of linear velocity and mass, **it uses their rotational motion counterparts: the object's moment of inertia and angular velocity.** The angular momentum of an object is denoted using the variable $\small L$ - The angular momentum of an object is measure in units of: $kg\cdot m^2/s$ $\newline$ $\text{Angular momentum: }\;\Huge\boxed{\color{LightCyan}\large L=I\omega}$ $\newline$ $\text{Angular impulse momentum theorem: }\;\Huge\boxed{\color{LightCyan}\large\Delta L =\tau\cdot\Delta t}$ $\newline$ - Based on the angular impulse momentum theorem equation we can see that the **change in the angular momentum of an object is proportional to the net torque applied to that object over some period of time.** - Just like with linear momentum, angular momentum also **abides by the concept of *conservation of momentum*.** That is; the **==total angular momentum of an isolated system will never change==**. This should be obvious based on the angular impulse momentum theorem since the equation implies that unless some external force *(or in this case torque)* acts upon a spinning object the momentum of that object will **never change**. - The *total angular momentum* is the vector sum of the momentum's of all the rotating objects that are *contained within a system.* - When an isolated rotating object experiences a change in its *moment of inertia* it will also experience a corresponding change in its angular velocity that **is inversely proportional to the change in the moment of inertia**. This relationship ensures that the angular momentum of the object stays constant. - *This is why you spin faster when you bring your arms closer to your body, lowering your moment of inertia...* $\newline$ $\Huge\boxed{\color{LightCyan}\large\text{Conservation of angular momentum: }\;L_f=L_i}$ $\newline$ --- ## Calculator <iframe src="https://app.calconic.com/api/embed/calculator/648919c428c7470029d3e720" sandbox="allow-same-origin allow-forms allow-scripts allow-top-navigation allow-popups-to-escape-sandbox allow-popups" title="Calconic_ Calculator" name="Calconic_ Calculator" height="8000" scrolling="no" style="width: 100%; border: 0; outline: none;"></iframe>