9.S: Linear Momentum and Collisions (Summary)
Key Terms
| center of mass | weighted average position of the mass |
| closed system | system for which the mass is constant and the net external force on the system is zero |
| elastic | collision that conserves kinetic energy |
| explosion | single object breaks up into multiple objects; kinetic energy is not conserved in explosions |
| external force | force applied to an extended object that changes the momentum of the extended object as a whole |
| impulse | effect of applying a force on a system for a time interval; this time interval is usually small, but does not have to be |
| impulse-momentum theorem | change of momentum of a system is equal to the impulse applied to the system |
| inelastic | collision that does not conserve kinetic energy |
| internal force | force that the simple particles that make up an extended object exert on each other. Internal forces can be attractive or repulsive |
| Law of Conservation of Momentum | total momentum of a closed system cannot change |
| linear mass density | \(\lambda\), expressed as the number of kilograms of material per meter |
| momentum | measure of the quantity of motion that an object has; it takes into account both how fast the object is moving, and its mass; specifically, it is the product of mass and velocity; it is a vector quantity |
| perfectly inelastic | collision after which all objects are motionless, the final kinetic energy is zero, and the loss of kinetic energy is a maximum |
| rocket equation | derived by the Soviet physicist Konstantin Tsiolkovsky in 1897, it gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass from m i down to m |
| system | object or collection of objects whose motion is currently under investigation; however, your system is defined at the start of the problem, you must keep that definition for the entire problem |
Key Equations
| Definition of momentum | $$\vec{p} = m \vec{v}$$ |
| Impulse | $$\vec{J} \equiv \int_{t_{i}}^{t_{f}} \vec{F} (t)dt\; or\; \vec{J} = \vec{F}_{ave} \Delta t$$ |
| Impulse-momentum theorem | $$\vec{J} = \Delta \vec{p}$$ |
| Average force from momentum | $$\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$$ |
| Instantaneous force from momentum (Newton’s second law) | $$\vec{F} (t) = \frac{d \vec{p}}{dt}$$ |
| Conservation of momentum | $$\frac{d \vec{p}_{1}}{dt} + \frac{d \vec{p}_{2}}{dt} = 0\; or\; \vec{p}_{1} + \vec{p}_{2} = constant$$ |
| Generalized conservation of momentum | $$\sum_{j = 1}^{N} \vec{p}_{j} = constant$$ |
| Conservation of momentum in two dimensions |
\[p_{f,x} = p_{1,i,x} + p_{2,i,x}\] \[p_{f,y} = p_{1,i,y} + p_{2,i,y}\] |
| External forces | $$\vec{F}_{ext} = \sum_{j = 1}^{N} \frac{d \vec{p}_{j}}{dt}$$ |
| Newton’s second law for an extended object | $$\vec{F} = \frac{d \vec{p}_{CM}}{dt}$$ |
| Acceleration of the center of mass | $$\vec{a}_{CM} = \frac{d^{2}}{dt^{2}} \left(\dfrac{1}{M} \sum_{j = 1}^{N} m_{j} \vec{r}_{j}\right) = \frac{1}{M} \sum_{j = 1}^{N} m_{j} \vec{a}_{j}$$ |
| Position of the center of mass for a system of particles | $$\vec{r}_{CM} \equiv \sum_{j = 1}^{N} m_{j} \vec{r}_{j}$$ |
| Velocity of the center of mass | $$\vec{v}_{CM} = \frac{d}{dt} \left(\dfrac{1}{M} \sum_{j = 1}^{N} m_{j} \vec{r}_{j}\right) = \frac{1}{M} \sum_{j = 1}^{N} m_{j} \vec{v}_{j}$$ |
| Position of the center of mass of a continuous object | $$\vec{r}_{CM} \equiv \frac{1}{M} \int \vec{r} dm$$ |
| Rocket equation | $$\Delta v = u \ln \left(\dfrac{m_{i}}{m}\right)$$ |
Summary
9.1 Linear Momentum
- The motion of an object depends on its mass as well as its velocity. Momentum is a concept that describes this. It is a useful and powerful concept, both computationally and theoretically. The SI unit for momentum is kg • m/s.
9.2 Impulse and Collisions
- When a force is applied on an object for some amount of time, the object experiences an impulse.
- This impulse is equal to the object’s change of momentum.
- Newton’s second law in terms of momentum states that the net force applied to a system equals the rate of change of the momentum that the force causes.
9.3 Conservation of Linear Momentum
- The law of conservation of momentum says that the momentum of a closed system is constant in time (conserved).
- A closed (or isolated) system is defined to be one for which the mass remains constant, and the net external force is zero.
- The total momentum of a system is conserved only when the system is closed.
9.4 Types of Collisions
- An elastic collision is one that conserves kinetic energy.
- An inelastic collision does not conserve kinetic energy.
- Momentum is conserved regardless of whether or not kinetic energy is conserved.
- Analysis of kinetic energy changes and conservation of momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one-dimensional, two-body collisions.
9.5 Collisions in Multiple Dimensions
- The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes.
- Momentum is conserved in both directions simultaneously and independently.
- The Pythagorean theorem gives the magnitude of the momentum vector using the x- and y-components, calculated using conservation of momentum in each direction.
9.6 Center of Mass
- An extended object (made up of many objects) has a defined position vector called the center of mass.
- The center of mass can be thought of, loosely, as the average location of the total mass of the object.
- The center of mass of an object traces out the trajectory dictated by Newton’s second law, due to the net external force.
- The internal forces within an extended object cannot alter the momentum of the extended object as a whole.
9.7 Rocket Propulsion
- A rocket is an example of conservation of momentum where the mass of the system is not constant, since the rocket ejects fuel to provide thrust.
- The rocket equation gives us the change of velocity that the rocket obtains from burning a mass of fuel that decreases the total rocket mass.