1. Chapter 6: Momentum and Collisions Momentum and Impulse  Linear momentum Linear momentum p of an object of mass m moving with velocity v is the product of its mass and velocity: SI unit: kilogram-meter per second (kg m/s)  Chang of momentum and force

2.  Momentum conservation Momentum and Impulse The (linear) momentum of an object is conserved when F net = 0.  Impulse If a constant force F acts on an object, the impulse I delivered to the object over a time interval  t is given by : SI unit: kilogram-meter per second (kg m/s) When a single constant force acts on an object, When the force is not constant, then Impulse-momentum theorem

3.  Impulse-momentum theorem Momentum and Impulse The impulse of the force acting on an object equals the change in momentum of that object as long as the time interval  t is taken to be arbitrarily small. An example of impulse The magnitude of the impulse delivered by a force during the time interval  t is equal to the area under the force vs. time graph or, equivalently, to F av  t.

4.  Examples Momentum and Impulse Example 6.1 : Teeing off A golf ball is struck with a club. The force on the ball varies from zero at contact and up to the max. value. (a) Find the impulse. m = 5.0x10 -2 kg v i = 0, v f = 44 m/s (b) Estimate the duration of the collision and the average force.

5.  Examples Momentum and impulse Example 6.2 : How good are the bumpers (a) Find the impulse delivered to the car. (b) Find the average force.  t=0.150 s

6.  Injury in automobile collisions Momentum and Impulse A force of about 90 kN compressing the tibia can cause fracture. Head accelerations of 150g experienced for about 4 ms or 50g for 60 ms are fatal 50% of the time. When the collision lasts for less than about 70 ms, a person will survive if the whole-body impact pressure (force per unit area) is less than 1.9x10 5 N/m 2. Death results in 50% of cases in which the whole-body impact pressure reaches 3.4x10 5 N/m 2. Consider a collision involving 75-kg passenger not wearing s seat belt, traveling at 27 m/s who comes to rest in 0.010 s after striking an unpadded dashboard. Fatal

7.  Conservation of momentum Conservation of Momentum average force on 1 by 2 average force on 2 by 1 Conservation of momentum When no net external force acts on a system, the total momentum of the system remains constant in time

8.  An example Conservation of Momentum Example 6.3 : The archer m 1 =60 kg (man + bow) m 2 =0.500 kg (arrow) speed of arrow v 2 =50.0 m/s The archer is moving opposite the direction of the arrow

9.  Three types of collisions Collisions Inelastic collision A collision in which momentum is conserved, but kinetic energy is not. Perfectly inelastic collision A collision between two objects in which both stick together after the collision. Elastic collision A collision in which both momentum and kinetic energy are conserved.

10.  Perfectly inelastic collisions Collisions Consider two objects with mass m 1 and m 2 moving with known initial velocities v 1i and v 2i along a straight line. They collide head-on and after the collision, they stick together and move with a common velocity v f.

11.  Examples of perfect inelastic collision Collisions Example 6.4 : An SUV vs. a compact m 1 =1.80x10 3 kg v 1i =15.0 m/s (a) Find the final speed after collision. m 2 =9.00x10 2 kg v 2i =-15.0 m/s (b) Find the changes in velocity. (c) Find the change in kinetic energy of the system.

12.  Examples of perfect inelastic collision Collisions Example 6.5 : Ballistic pendulum m 1 =5.00 g m 2 =1.00 kg h = 5.00 cm Find the initial speed of bullet. At the height h Right after collisionBefore collision Right after collision

13.  Elastic collisions Collisions Consider two objects with mass m 1 and m 2 moving with known initial velocities v 1i and v 2i along a straight line. They collide head-on and after the collision, they leave each other with velocities v 1f and v 2f.

14.  An example of elastic collision Collisions Example 6.7 : Two blocks and a spring (a) Find v 2f when v 1f =+3.00 m/s. m 2 =2.10 kg v 2i =-2.50 m/s (b) Find the compression of the spring. m 1 =1.60 kg v 1i =+4.00 m/s k=6.00x10 2 N/m

15.  Collisions in 2-dimension Glancing Collisions Momentum conservation in 2-D

16.  An example of a collision in 2-D Glancing Collisions Example 6.8 : A perfect inelastic collision at an intersection m car =1.50x10 3 kg m van =2.50x10 3 kg Find the magnitude and direction of the velocity of the wreckage.

17.  An example of a collision in 2-D (cont’d) Glancing Collisions Example 6.8 : A perfect inelastic collision at an intersection (cont’d) m car =1.50x10 3 kg m van =2.50x10 3 kg Find the magnitude and direction of the velocity of the wreckage.

18.  Principle (hand-waving argument) Rocket Propulsion The driving force of motion of ordinary vehicles such as cars and locomotives is friction. A car moves because a reaction to the force exerted by the tire produces a force by the road on the wheel. What is then driving force of a rocket?  When an explosion occurs in a spherical chamber with fuel gas in a rocket engine the hot gas expands and presses against all sides of the chamber uniformly. So all forces are in balance-no net force.  If there is a hole as in (b), part of the hot gas escapes from the hole (nozzle), which breaks the balance of the forces. This unbalance create a net upward force.

19.  Principle (detailed argument) Rocket Propulsion At time t, the momentum of the rocket plus the fuel is (M+  m)v. M : mass of rocket  m : mass of fuel to be ejected in  t During time period  t, the rocket ejects fuel of mass  m whose speed v e relative to the rocket and gains the speed to v+  v. From momentum conservation: The increase  m in the mass of the exhaust corresponds to an equal decrease in the mass of the rocket so that  m=-  M.

20.  Principle (detailed argument) Rocket Propulsion Using calculus: M : mass of rocket  m : mass of fuel to be ejected in  t  Thrust is defined as the force exerted on the rocket by the ejected exhaust gases. Instantaneous thrust

21.  An example Rocket Propulsion Example 6.0 : Single stage to orbit M : mass of rocket 1.00x10 5 kg m : burnout mass 1.00x10 4 kg v e : exhaust velocity 4.50x10 3 m/s  t : blast off time period 4 min (b) Find the thrust at liftoff. Thrust (c) Find the initial acceleration.