Momentum and Collisions Chapter 6 Momentum and Collisions

Momentum The linear momentum of an object of mass m moving with a velocity is defined as the product of the mass and the velocity SI Units are kg m / s Vector quantity, the direction of the momentum is the same as the velocity’s

Momentum components Applies to two-dimensional motion

Impulse In order to change the momentum of an object, a force must be applied The time rate of change of momentum of an object is equal to the net force acting on it Gives an alternative statement of Newton’s second law

Impulse cont. When a single, constant force acts on the object, there is an impulse delivered to the object is defined as the impulse Vector quantity, the direction is the same as the direction of the force

Impulse-Momentum Theorem The theorem states that the impulse acting on the object is equal to the change in momentum of the object If the force is not constant, use the average force applied

Impulse Applied to Auto Collisions The most important factor is the collision time or the time it takes the person to come to a rest This will reduce the chance of dying in a car crash Ways to increase the time Seat belts Air bags

Air Bags The air bag increases the time of the collision It will also absorb some of the energy from the body It will spread out the area of contact decreases the pressure helps prevent penetration wounds

Example 1 Calculate the magnitude of the linear momentum for the following cases: (a) a proton with mass 1.67 × 10–27 kg, moving with a speed of 5.00 × 106 m/s; (b) a 15.0-g bullet moving with a speed of 300 m/s; (c) a 75.0-kg sprinter running with a speed of 10.0 m/s; (d) the Earth (mass = 5.98 × 1024 kg) moving with an orbital speed equal to 2.98 × 104 m/s.

Example 2 A 0.10-kg ball is thrown straight up into the air with an initial speed of 15 m/s. Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height.

Example 3 A 75.0-kg stuntman jumps from a balcony and falls 25.0 m before colliding with a pile of mattresses. If the mattresses are compressed 1.00 m before he is brought to rest, what is the average force exerted by the mattresses on the stuntman?

Example 4 A 0.500-kg football is thrown toward the east with a speed of 15.0 m/s. A stationary receiver catches the ball and brings it to rest in 0.020 0 s. (a) What is the impulse delivered to the ball as it’s caught? (b) What is the average force exerted on the receiver?

Example 5 A force of magnitude Fx acting in the x-direction on a 2.00-kg particle varies in time as shown in Figure P6.12. Find (a) the impulse of the force, (b) the final velocity of the particle if it is initially at rest, and (c) the final velocity of the particle if it is initially moving along the x-axis with a velocity of –2.00 m/s.

Example 6 A pitcher throws a 0.15-kg baseball so that it crosses home plate horizontally with a speed of 20 m/s. The ball is hit straight back at the pitcher with a final speed of 22 m/s. (a) What is the impulse delivered to the ball? (b) Find the average force exerted by the bat on the ball if the two are in contact for 2.0 × 10–3 s.

Conservation of Momentum Momentum in an isolated system in which a collision occurs is conserved A collision may be the result of physical contact between two objects “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies An isolated system will have not external forces

Conservation of Momentum, cont The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system remains constant in time Specifically, the total momentum before the collision will equal the total momentum after the collision

Conservation of Momentum, cont. Mathematically: Momentum is conserved for the system of objects The system includes all the objects interacting with each other Assumes only internal forces are acting during the collision Can be generalized to any number of objects

Notes About A System Remember conservation of momentum applies to the system You must define the isolated system

Types of Collisions Momentum is conserved in any collision Inelastic collisions Kinetic energy is not conserved Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object Perfectly inelastic collisions occur when the objects stick together Not all of the KE is necessarily lost

More Types of Collisions Elastic collision both momentum and kinetic energy are conserved Actual collisions Most collisions fall between elastic and perfectly inelastic collisions

More About Perfectly Inelastic Collisions When two objects stick together after the collision, they have undergone a perfectly inelastic collision Conservation of momentum becomes

Some General Notes About Collisions Momentum is a vector quantity Direction is important Be sure to have the correct signs

More About Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns Solve the equations simultaneously

Summary of Types of Collisions In an elastic collision, both momentum and kinetic energy are conserved In an inelastic collision, momentum is conserved but kinetic energy is not In a perfectly inelastic collision, momentum is conserved, kinetic energy is not, and the two objects stick together after the collision, so their final velocities are the same

Problem Solving for One -Dimensional Collisions Coordinates: Set up a coordinate axis and define the velocities with respect to this axis It is convenient to make your axis coincide with one of the initial velocities Diagram: In your sketch, draw all the velocity vectors and label the velocities and the masses

Problem Solving for One -Dimensional Collisions, 2 Conservation of Momentum: Write a general expression for the total momentum of the system before and after the collision Equate the two total momentum expressions Fill in the known values

Problem Solving for One -Dimensional Collisions, 3 Conservation of Energy: If the collision is elastic, write a second equation for conservation of KE, or the alternative equation This only applies to perfectly elastic collisions Solve: the resulting equations simultaneously

Sketches for Collision Problems Draw “before” and “after” sketches Label each object include the direction of velocity keep track of subscripts

Sketches for Perfectly Inelastic Collisions The objects stick together Include all the velocity directions The “after” collision combines the masses

Glancing Collisions For a general collision of two objects in three-dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved Use subscripts for identifying the object, initial and final velocities, and components

Glancing Collisions The “after” velocities have x and y components Momentum is conserved in the x direction and in the y direction Apply conservation of momentum separately to each direction

Problem Solving for Two-Dimensional Collisions Coordinates: Set up coordinate axes and define your velocities with respect to these axes It is convenient to choose the x- or y- axis to coincide with one of the initial velocities Draw: In your sketch, draw and label all the velocities and masses

Problem Solving for Two-Dimensional Collisions, 2 Conservation of Momentum: Write expressions for the x and y components of the momentum of each object before and after the collision Write expressions for the total momentum before and after the collision in the x-direction and in the y-direction

Problem Solving for Two-Dimensional Collisions, 3 Conservation of Energy: If the collision is elastic, write an expression for the total energy before and after the collision Equate the two expressions Fill in the known values Solve the quadratic equations Can’t be simplified

Problem Solving for Two-Dimensional Collisions, 4 Solve for the unknown quantities Solve the equations simultaneously There will be two equations for inelastic collisions There will be three equations for elastic collisions

Example 7 A rifle with a weight of 30 N fires a 5.0-g bullet with a speed of 300 m/s. (a) Find the recoil speed of the rifle. (b) If a 700-N man holds the rifle firmly against his shoulder, find the recoil speed of the man and rifle.

Example 8 A 75.0-kg ice skater moving at 10.0 m/s crashes into a stationary skater of equal mass. After the collision, the two skaters move as a unit at 5.00 m/s. Suppose the average force a skater can experience without breaking a bone is 4 500 N. If the impact time is 0.100 s, does a bone break?

Example 9 A 0.030-kg bullet is fired vertically at 200 m/s into a 0.15-kg baseball that is initially at rest. How high does the combined bullet and baseball rise after the collision, assuming the bullet embeds itself in the ball?

Example 10 A 1 200-kg car traveling initially with a speed of 25.0 m/s in an easterly direction crashes into the rear end of a 9 000-kg truck moving in the same direction at 20.0 m/s (Fig. P6.32). The velocity of the car right after the collision is 18.0 m/s to the east. (a) What is the velocity of the truck right after the collision? (b) How much mechanical energy is lost in the collision? Account for this loss in energy.

Example 11 A 12.0-g bullet is fired horizontally into a 100-g wooden block that is initially at rest on a frictionless horizontal surface and connected to a spring having spring constant 150 N/m. The bullet becomes embedded in the block. If the bullet–block system compresses the spring by a maximum of 80.0 cm, what was the speed of the bullet at impact with the block?

Example 12 A billiard ball rolling across a table at 1.50 m/s makes a head-on elastic collision with an identical ball. Find the speed of each ball after the collision (a) when the second ball is initially at rest, (b) when the second ball is moving toward the first at a speed of 1.00 m/s, and (c) when the second ball is moving away from the first at a speed of 1.00 m/s.

Example 13 A 90-kg fullback moving east with a speed of 5.0 m/s is tackled by a 95-kg opponent running north at 3.0 m/s. If the collision is perfectly inelastic, calculate (a) the velocity of the players just after the tackle and (b) the kinetic energy lost as a result of the collision. Can you account for the missing energy?