Momentum and Impulse: Understanding Collisions

1. Chapter 7 Impulse and Momentum

2. 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

3. Momentum components • Applies to two-dimensional motion

4. 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, e.g. • Gives an alternative statement of Newton’s second law

5. 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 • Unit N·s=kg·m/s

6. Impulse-Momentum Theorem • The theorem states that the impulse acting on the object is equal to the change in momentum of the object • Impulse=change in momentum (vector!) • If the force is not constant, use the average force applied

7. 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

8. 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

9. Example 0.05 kg ball moving at 2.0 m/s rebounds with the same speed. If the contact time with the wall is 0.01 s, what is average force of the wall on the ball?

10. Conservation of Momentum • Total momentum of a system equals to the vector sum of the momenta • When no resultant external force acts on a system, the total momentum of the system remains constant in magnitude and direction. • Components of Momentum

11. 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

12. 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

13. 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

14. Example Two skaters are initially at rest. Masses are 80kg and 50kg. If they push each other so that woman is given a velocity of 2.5 m/s. What is the velocity of the man?

15. Types of Collisions • Momentum is conserved in any collision • Perfect elastic collision • both momentum and kinetic energy are conserved • Collision of billiard balls, steel balls

16. More Types of Collisions • 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 • completely inelastic collisions occur when the objects stick together • Not all of the KE is necessarily lost • Actual collisions • Most collisions fall between elastic and completely inelastic collisions

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

18. Example Railroad car (10,000kg) travels at 10m/s and strikes another railroad car (15,000kg) at rest. They couple after collision. Find the final velocity of the two cars. What is the energy loss in the collision?

19. Recoil • System is released from rest • Momentum of the system is zero before and after

20. Example 4 kg rifle shoots a 50 grams bullet. If the velocity of the bullet is 280 m/s, what is the recoil velocity of the rifle?

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

22. More About Elastic Collisions • Both momentum and kinetic energy are conserved • Typically have two unknowns (1d) • Solve the equations simultaneously

23. A Simple Case, v2i=0 • Head on elastic collision with object 2 at rest before collision. • One can show • Special cases

25. Ballistic Pendulum • Measure speed of bullet • Momentum conservation of the collision • Energy conservation during the swing of the pendulum

26. 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

27. Example 7.31. Balls A and B collide head-on in a perfectly elastic collision. It is known that mA=2mB and that the initial velocities are +3 m/s for A and –2 m/s for B. Find their velocities after the collision.

28. 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

30. Example • Car, 1500 kg. SUV 2500 kg. • Find speed and direction after collision.

31. Example m1=0.15 kg, m2=0.26 kg, v1i=0.9 m/s at 50° to y-axis, v2i=0.54 m/s, v2f=0.7 m/s at 35° below x-axis Find v2f (magnitude and direction)