Chapter 9 Linear Momentum and Collisions Linear Momentum The linear momentum of a particle or an object that can be modeled as a particle of mass m moving with a velocity v is defined to be the product of the mass and velocity: p = m v The terms momentum and linear momentum will be used interchangeably in the text 1
Linear Momentum, cont Linear momentum is a vector quantity Its direction is the same as the direction of v The dimensions of momentum are ML/T The SI units of momentum are kg m / s Momentum can be expressed in component form: p x = m v x p y = m v y p z = m v z Newton and Momentum Newton called the product mv the quantity of motion of the particle Newton s Second Law can be used to relate the momentum of a particle to the resultant force acting on it dv d( mv) dp Σ F = ma= m = = dt dt dt with constant mass 2
Conservation of Linear Momentum Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant The momentum of the system is conserved, not necessarily the momentum of an individual particle This also tells us that the total momentum of an isolated system equals its initial momentum Conservation of Momentum, 2 Conservation of momentum can be expressed mathematically in various ways p total = p 1 + p 2 = constant p 1i + p 2i = p 1f + p 2f In component form, the total momenta in each direction are independently conserved p ix = p fx p iy = p fy p iz = p fz Conservation of momentum can be applied to systems with any number of particles 3
Conservation of Momentum, Archer Example The archer is standing on a frictionless surface (ice) Approaches: Newton s Second Law no, no information about F or a Energy approach no, no information about work or energy Momentum yes Conservation of Momentum, Example A honeybee with a mass of 0.175 g lands on one end of a popsicle stick. After sitting at rest for a moment, the bee runs toward the other end with a velocity 1.25 cm/s relative to the still water. What is the speed of the 4.95 g stick relative to the water? (Assume the bee's motion is in the negative direction.) ans:0.442 mm/s 4
Impulse and Momentum From Newton s Second Law, F = dp/dt Solving for dp gives dp = Fdt Integrating to find the change in momentum over some time interval f = f i = dt = ti p p p F I The integral is called the impulse, I, of the force F acting on an object over t t Impulse-Momentum Theorem This equation expresses the impulsemomentum theorem:the impulse of the force F acting on a particle equals the change in the momentum of the particle This is equivalent to Newton s Second Law 5
Impulse-Momentum: Crash Test Example The momenta before and after the collision between the car and the wall can be determined (p = m v) Find the impulse: I = p = p f p i F = p / t Impulse-Momentum: TENNIS A tennis player receives a shot with the ball (0.0600 kg) traveling horizontally at 54.0 m/s and returns the shot with the ball traveling horizontally at 42.0 m/s in the opposite direction. (Assume the initial direction of the ball is in the -x direction.) (a) What is the impulse delivered to the ball by the racquet? 5.76 kg m/s in the +x direction (b) What work does the racquet do on the ball? -34.6 J 6
Collisions Characteristics We use the term collision to represent an event during which two particles come close to each other and interact by means of forces The time interval during which the velocity changes from its initial to final values is assumed to be short The interaction force is assumed to be much greater than any external forces present This means the impulse approximation can be used Types of Collisions In an elastic collision, momentum and kinetic energy are conserved Perfectly elastic collisions occur on a microscopic level In macroscopic collisions, only approximately elastic collisions actually occur In an inelastic collision, kinetic energy is not conserved although momentum is still conserved If the objects stick together after the collision, it is a perfectly inelastic collision 7
Perfectly Inelastic Collisions Since the objects stick together, they share the same velocity after the collision m 1 v 1i + m 2 v 2i = (m 1 + m 2 ) v f Elastic Collisions Both momentum and kinetic energy are conserved m v + m v = 1 1i 2 2i mv + m v 1 1f 2 2 f 1 2 1 2 m1v1i + m2v2i = 2 2 1 1 mv + m v 2 2 2 2 1 1f 2 2 f 8
Elastic Collisions, cont Typically, there are two unknowns to solve for and so you need two equations The kinetic energy equation can be difficult to use With some algebraic manipulation, a different equation can be used v 1i v 2i = v 1f + v 2f This equation, along with conservation of momentum, can be used to solve for the two unknowns It can only be used with a one-dimensional, elastic collision between two objects The Center of Mass There is a special point in a system or object, called the center of mass, that moves as if all of the mass of the system is concentrated at that point The system will move as if an external force were applied to a single particle of mass M located at the center of mass M is the total mass of the system 9
Center of Mass, Coordinates The coordinates of the center of mass are mx i i my i i mz i i i i i xcm = ycm = zcm = M M M where M is the total mass of the system Center of Mass, position The center of mass can be located by its position vector, r CM r CM m r i i i = M r i is the position of the i th particle, defined by r = xiˆ+ yˆj+ z kˆ i i i i 10
Center of Mass, Example Both masses are on the x-axis The center of mass is on the x-axis The center of mass is closer to the particle with the larger mass 11