Think-Pair-Share. Linear Momentum (Ch 9) Linear Momentum, cont. Newton and Momentum

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1 Linear Momentum (Ch 9) 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 Think-Pair-Share What is the magnitude of the momentum of a 28-g sparrow flying with a speed of 8.4 m/s? 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: 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 p x = m v x p y = m v y p z = m v z with constant mass 1

2 Newton s Second Law The time rate of change of the linear momentum of a particle is equal to the net force acting on the particle This is the form in which Newton presented the Second Law It is a more general form than the one we used previously This form also allows for mass changes Applications to systems of particles are particularly powerful 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 Think-Pair-Share A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind. When he was tackled by an 85-kg cornerback running at 5.5 m/s in the same direction, what was their mutual speed immediately after the tackle? 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 2

3 Conservation of Momentum, Kaon Example The kaon decays into a positive π and a negative π particle Total momentum before decay is zero Therefore, the total momentum after the decay must equal zero p + + p - = 0 or p + = -p - Think-Pair-Share A radioactive nucleus at rest decays into a second nucleus, an electron, and a neutrino. The electron and neutrino are emitted at right angles and have momenta of 9.30 x kg m/s and 5.40 x kg m/s, respectively. What are the magnitude and direction of the momentum of the second recoiling nucleus? 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 The integral is called the impulse, I, of the force F acting on an object over Δt More About Impulse Impulse is a vector quantity The magnitude of the impulse is equal to the area under the force-time curve Dimensions of impulse are M L / T Impulse is not a property of the particle, but a measure of the change in momentum of the particle 3

4 Impulse, Final Ch 9: Problem 7 The impulse can also be found by using the time averaged force I = Δt This would give the same impulse as the time-varying force does An estimated force-time curve for a baseball struck by a bat is shown. From this curve, determine (a) the impulse delivered to the ball, (b) the average force exerted on the ball, and (c) the peak force exerted on the ball. 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 = Δp = p f p i F = Δp / Δt Collisions Example The collision need not include physical contact between the objects There are still forces between the particles This type of collision can be analyzed in the same way as those that include physical contact 4

5 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 Momentum is conserved in all collisions 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 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 5

6 Elastic Collisions, final Example of some special cases m 1 = m 2 the particles exchange velocities When a very heavy particle collides head-on with a very light one initially at rest, the heavy particle continues in motion unaltered and the light particle rebounds with a speed of about twice the initial speed of the heavy particle When a very light particle collides head-on with a very heavy particle initially at rest, the light particle has its velocity reversed and the heavy particle remains approximately at rest Collision Example Ballistic Pendulum Perfectly inelastic collision the bullet is embedded in the block of wood Momentum equation will have two unknowns Use conservation of energy from the pendulum to find the velocity just after the collision Then you can find the speed of the bullet Ballistic Pendulum, cont A multi-flash photograph of a ballistic pendulum Ch 9: Problem 17 A 10.0-kg bullet is fired into a stationary block of wood (m=5.00 km). The relative motion of the bullet stops inside the block. The speed of the bullet-plus-block combination immediately after the collision is m/s. What was the original speed of the bullet? 6

7 Two-Dimensional Collisions The momentum is conserved in all directions Use subscripts for identifying the object indicating initial or final values the velocity components If the collision is elastic, use conservation of kinetic energy as a second equation Remember, the simpler equation can only be used for one-dimensional situations Two-Dimensional Collision, example Particle 1 is moving at velocity v 1i and particle 2 is at rest In the x-direction, the initial momentum is m 1 v 1i In the y-direction, the initial momentum is 0 Two-Dimensional Collision, example cont After the collision, the momentum in the x- direction is m 1 v 1f cos θ + m 2 v 2f cos φ After the collision, the momentum in the y- direction is m 1 v 1f sin θ + m 2 v 2f sin φ Analysis of Momentum Momentum is a vector. x and y components need to be analyzed independently. 7

8 Two-Dimensional Collision Example Before the collision, the car has the total momentum in the x- direction and the van has the total momentum in the y-direction After the collision, both have x- and y- components Ch 9: Problem 29 Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed v i =5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37 0 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk. 8

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