Fundamentals of Physics School of Physical Science and Technology Mechanics (Bilingual Teaching) 张昆实 School of Physical Science and Technology Yangtze University

Chapter 10 Collisions 10-1 What is a Collision? 10-2 Impulse and Linear Momentum 10-3 Momentum and Kinetic Energy in Collisions 10-4 Inelastic Collisions in One Dimension 10-5 Elastic Collisions In One 10-6 Collisions in Two Dimensions

10-1 What is a Collision? In everyday language, a collision occurs when an objects crash into each other. Exp: (a) Meteor Crater (流星坑) (b) particles scattering (粒子散射) (c) tennis ball contacts with racket A collision is an isolated event in which two or more bodies (the colliding bodies) exert relatively strong forces on each other for a relatively short time. definition Note : a collision does not require contact, a collision force does not have to be a force involving contact.

10-2 Impulse and Linear Momentum Single collision During a head-on collision: A third law force pair , The change of the linear momentum depends on: The forces and the action time Apply Newton’s second law to ball R (10-1) Integrating Eq. 10-1 over the interval : (10-2)

10-2 Impulse and Linear Momentum (10-2) (10-3) is the impulse, which is a measure of both the strength and the duration of the collision force. J rectangle (10-8)

10-2 Impulse and Linear Momentum (10-2) (10-3) From Eq. 10-2 and 10-3 (10-4) ( Impulse-linear momentum theorem ) Component form (10-5) (10-6) (10-7)

10-2 Impulse and Linear Momentum Series of collisions Projectiles Target A steady stream of projectile bodies ( ), moves along axis and collides with a fixed target. Find the average force on the target during the bombardment. The total change in momentum for projectiles in is : The impulse on the target in : (10-9) Combining Eq. 10-8 and 10-9: (10-10)

10-2 Impulse and Linear Momentum Series of collisions Projectiles Target The rate the projectiles collide with the target (10-10) In , an amount of mass collide with the target, so (10-13) The rate the mass collides with the target If If

10-3 Momentum and Kinetic Energy in Collisions discussion collisions in closed (no mass enters or leaves them) and isolated (no net external forces act on the bodies within them ) systems. Kinatic Energy（in collisions） 1. Elastic collision: is a special type of collision in which the kinetic energe of the system of colliding bodies is conserved. 2. Inelastic collision: after the collision the kinetic energe of the system is not conserved. 3. Completely inelastic collision: If the bodies stick together and have the same final velocity after the collision.

10-3 Momentum and Kinetic Energy in Collisions Linear Momentum（in collisions） Regardless of the details of the impulses in a collision; Regardless of what happens to the kinetic energy of the system, The total linear momentum of a closed, isolated systems cannot change. (no external force !) The law of conservation of linear momentum In a closed , isolated system containing a collision, the linear momentum of each colliding body may change but the total linear momentum of the system cannot change, whether the collision is elastic or inelastic.

10-4 Inelastic Collisions in One Dimension befor after One dimensional collision Two colliding bodies form a closed , isolated system along the x axis. The law of conservation of linear momentum (10-15) before、 after collision (10-16) The motion is one-dimensional, component form:

10-4 Inelastic Collisions in One Dimension Completely Inelastic collision after collision befor Projectile Target Before the collision body 2 is at rest, body 1 moves directly toward it. After the collision, the stuck-together bodies move with the same velocity . The law of conservation of linear momentum (10-17) before、 after collision (10-18)

10-4 Inelastic Collisions in One Dimension Velocity of Center of Mass Collision ! In a closed, isolated system, the velosity of the center of mass of the system cannot be changed by a collision for there is no net external force to change it. Find out (10-19) (10-20) Example (10-21)

10-5 Elastic Collisions In One Dimension after befor Projectile Target Stationary Target An elastic collision is a special type of collision in which the kinetic energy of the system of cilliding bodies is conserved. Total kinetic energy before the collision after the collision = ☞ In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.

10-5 Elastic Collisions In One Dimension after befor Projectile Target Stationary Target Before the collision body 2 is at rest, body 1 moves toward it (head on collision). Assume this two body system is closed and Isolated. Then the net linear momentum of the system is conserved. ( linear momentum ) (10-26) ☞ If the collision is also elastic, the total kinetic energy of the system is also conserved. (10-27) ( kinetic energy )

10-5 Elastic Collisions In One Dimension Stationary Target ( linear momentum ) (10-26) (10-27) ( kinetic energy ) If know , Then can find out (10-28) Rewrite Eq.10-26 (10-29) Rewrite Eq.10-27 (10-30) (10-31)

10-5 Elastic Collisions In One Dimension after befor Projectile Target Stationary Target (10-30) (10-31) discussion 1. From (10-30) , ball 2 always moves forward 2. From (10-31), if ball 1 moves forward ball 1 bounds back If

10-5 Elastic Collisions In One Dimension Stationary Target (10-30) (10-31) discussion A few special situations 1. Equal masses: If Eqs. (10-30) and (10-31) reduce to and Changing vilocities ! 2. A massive target: If Eqs. (10-30) and (10-31) reduce to and (10-32) bounds back 3. A massive projectile: If Eqs. (10-30) and (10-31) reduce to and (10-33)

10-5 Elastic Collisions In One Dimension 1.Conservation of the linear momentum Moving Target 2.Conservation of the kinetic energy head on elastic collision (10-36) To work out and rewrite these conservation equations: (10-37) (10-38) (10-39)

10-6 Collisions in Two Dimensions glancing collision Two dimension (not head-on) elastic collision System: closed + isolated Conservation equations: (10-40) (10-41) Rewrite Eq.10-40 for components along the x、y axis (10-42) (10-43) (10-44)