9.S: Linear Momentum and Collisions (Summary)
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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 | λ, 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 mi 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 | →p=m→v |
Impulse | →J≡∫tfti→F(t)dtor→J=→FaveΔt |
Impulse-momentum theorem | →J=Δ→p |
Average force from momentum | →F=Δ→pΔt |
Instantaneous force from momentum (Newton’s second law) | →F(t)=d→pdt |
Conservation of momentum | d→p1dt+d→p2dt=0or→p1+→p2=constant |
Generalized conservation of momentum | N∑j=1→pj=constant |
Conservation of momentum in two dimensions |
pf,x=p1,i,x+p2,i,x pf,y=p1,i,y+p2,i,y |
External forces | →Fext=N∑j=1d→pjdt |
Newton’s second law for an extended object | →F=d→pCMdt |
Acceleration of the center of mass | →aCM=d2dt2(1MN∑j=1mj→rj)=1MN∑j=1mj→aj |
Position of the center of mass for a system of particles | →rCM≡N∑j=1mj→rj |
Velocity of the center of mass | →vCM=ddt(1MN∑j=1mj→rj)=1MN∑j=1mj→vj |
Position of the center of mass of a continuous object | →rCM≡1M∫→rdm |
Rocket equation | Δv=uln(mim) |
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.