2 What you have to know Momentum and impulse Momentum conservation Various collision problems –The typical examples where momentum conservation is used are collisions! Center of mass Section 8.6 is not in the curriculum.
3 Newton’s 2 nd law of motion This cannot be applied to the problems where mass m is changing. E.g. Rocket propulsion problem. (See Sec 9.6) Original form of the 2 nd law of motion written by Newton Momentum is a vector quantity.
4 The impulse-momentum theorem Impulse Impulse is a vector quantity. Impulse-momentum theorem The change in momentum of a particle during a time interval equals the impulse of the net force.
5 Impulse General definitionImpulse-momentum theorem Impulse: area under the curve of net force in F-t graph Same area Same impulse Large force for a short time Smalle force for a longer time
6 Momentum vs. Kinetic Energy Integral form of Newton’s 2 nd law Differential form of Newton’s 2 nd law identical
7 Ex 8.2A ball hits a wall before after
8 Ex 8.3 Before After This is a 2-dim. motion. First decompose the momentum into x and y directions
9 Conservation of Momentum Internal forces External force Isolated system: External force = 0 Newton’s 2 nd law Newton’s 3 rd law You can add these forces to obtain the net (internal) force for the system The sum of these forces cannot be applied to a single object.
10 Conservation of Momentum Total momentum Total momentum of an isolated system is conserved. Momentum conservation gives a vector equation.
11 Ex 8.4 Recoil of a rifle Before After Rifle + bullet Riflebullet Note that momentum conservation is valid for an isolated system. So it is useful when the system changes. Therefore, draw diagrams for “before” & “after” the event change the unit so that the both masses have the same unit The negative sign means that the rifle is moving back!
12 Ex 8.4 (cont’d) The sum of these two momenta vanishes as it should be. The kinetic energy is NOT conserved. The kinetic energy before the event was 0.
13 Ex 8.5 Collision along a straight line AB AB Before After If you have a collision problem, always use momentum conservation.
14 Ex 8.6 Collision in a horizontal plane Before After Again, we use momentum conservation. Also note that momentum is a vector quantity. Use the numbers above
15 Momentum conservation and collisions The key to solve collision problems is Momentum conservation. 1.At the instant of collision, very complex and very large forces are acting on the bodies. 2.Then the other forces can be neglected and the system is almost an isolated system. momentum conservation can be applied 3.The forces are very complex and we do not know much about the forces. momentum conservation can give useful information for the motions of the colliding bodies Elastic collision If the forces between the colliding bodies are conservative, no mechanical energy is lost. The total kinetic energy of the system is conserved. For elastic collisions, the total momentum and the total kinetic energy of the system are conserved.
16 Inelastic collision In this collision, the total kinetic energy of the system after the collision is less than before the collision. The forces during the collision are non-conservative. Completely inelastic collision This is a special case of inelastic collisions. This is where the colliding bodies stick together and move as one body after the collision.
17 Completely inelastic collisions What happens to momentum and kinetic energy in a completely inelastic collision of two bodies AB AB Before After Consider a 1-dim collision where v B1 = 0
18 Ex 8.8 The ballistic pendulum This is a device which can measure the speed of a bullet. Before collision Immediately after collision Swing of the body after collision Analyze this event in two stages ! Collision process One-body motion This gives a relation between v 1 and v 2. This gives a relation between v 2 and the height y. (A)(B) So, if you know y, you can know v 2 and v 1
19 Ex 8.8 (cont’d) The first stage: (A) collision The second stage: (B) one-body motion Combining the two By measuring the height y, you can know the speed of the bullet
20 Ex 8.9 Automobile collision Before After Completely inelastic collision
21 Elastic collisions Consider 1-dim elastic collision. AB AB Before After These give TWO equations. You now have TWO unknowns (the final velocities). Therefore, you can completely determine the unknowns.
22 Elastic collisions: one body initially at rest rearrange AB AB Before After
23 A B A Before After B A B B Before After A 1.The motion of the heavy object doesn’t change. 2.The light object moves to the opposite direction but with same speed. 1.The motion of the heavy object doesn’t change. 2.The light object moves in the direction of the heavy one but (two times) faster than the heavy one.
24 Elastic collisions & relative velocity The velocity of B relative to A after the collision The negative of the velocity of B relative to A before the collision If you watch the motion of A sitting on B, then you will see: A is approaching to B with speed v before the collision. After the collision, A is moving away from B with the same speed
25 Ex 8.12 Two-dim elastic collision (If this is a head-on collision, it is the same as the 1-dim collision.) Before After FOUR unknowns after collision (two velocity vectors) Momentum conservation gives two eqs. Kinetic energy conservation gives one eq. So totally THREE equations. To uniquely determine the final velocities, we need ONE more information for the final velocities
26 Ex 8.12 (cont’d)
27 Center of mass We now introduce the concept of center of mass (CM) i-th particle
28 Motion of the center of mass Total momentum of the system Momentum of the CM You can represent an extended object as a point- like particle; the total mass of the object is at its Center of Mass (CM)
29 Motion of the CM ` When a shell explodes This is the trajectory of the shell if it doesn’t explode.