Momentum Conservation

Conservation of Linear Momentum The net force acting on an object is the rate of change of its momentum: If the net force is zero, the momentum does not change:

Conservation of Linear Momentum Furthermore, internal forces cannot change the momentum of a system. However, the momenta of components of the system may change.

1. vA > vC 2. vA = vC 3. vA < vC Objects A and C are made of different materials, with different “springiness,” but they have the same mass and are initially at rest. When ball B collides with object A, the ball ends up at rest. When ball B is thrown with the same speed and collides with object C, the ball rebounds to the left. Compare the velocities of A and C after the collisions. Is vA greater than, equal to, or less than vC? 1. vA > vC 2. vA = vC 3. vA < vC

1. vA > vC 2. vA = vC 3. vA < vC Objects A and C are made of different materials, with different “springiness,” but they have the same mass and are initially at rest. When ball B collides with object A, the ball ends up at rest. When ball B is thrown with the same speed and collides with object C, the ball rebounds to the left. Compare the velocities of A and C after the collisions. Is vA greater than, equal to, or less than vC? 1. vA > vC 2. vA = vC 3. vA < vC

Conservation of Linear Momentum An example of internal forces moving components of a system:

Canoe Example Net momentum = 0 Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in Canoe 1 (total mass 130 kg) pushes on Canoe 2 (total mass 250 kg) with a force of 46 N to separate the canoes. Find the momentum of each canoe after 1.20 s of pushing. Net momentum = 0

Example: Velocity of a Bee A honeybee with a mass of 0.150 g lands on one end of a floating 4.75 g popsicle stick. After sitting at rest for a moment, it runs to the other end of the stick with a velocity vb relative to still water. The stick moves in the opposite direction with a velocity of 0.120 cm/s. Find the velocity vb of the bee.

Elastic & Inelastic Collisions Collision: two objects striking one another. Time of collision is short enough that external forces may be ignored. Elastic collision: both momentum & kinetic energy are conserved Inelastic collision: momentum is conserved but kinetic energy is not: pf = pi but Kf ≠ Ki. Completely inelastic collision: objects stick together afterwards: pf = pi1 + pi2

Inelastic Collisions A completely inelastic collision:

Example: Goal-Line Stand On a touchdown attempt, a 95.0 kg running back runs toward the end zone at 3.75 m/s. A 111 kg line backer moving at 4.10 m/s meets the runner in a head-on collision and locks his arms around the runner. (a) Find their velocity immediately after the collisions. (b) Find the initial and final kinetic energies and the energy DK lost in the collision.

The two particles are both moving to the right The two particles are both moving to the right. Particle 1 catches up with particle 2 and collides with it. The particles stick together and continue on with velocity vf. Which of these statements is true? 1. vf is greater than v1. 2. vf = v1. 3. vf is less than v2. 4. vf = v2. 5. vf is greater than v2, but less than v1.

The two particles are both moving to the right The two particles are both moving to the right. Particle 1 catches up with particle 2 and collides with it. The particles stick together and continue on with velocity vf. Which of these statements is true? 1. vf is greater than v1. 2. vf = v1. 3. vf is less than v2. 4. vf = v2. 5. vf is greater than v2, but less than v1.

Inelastic Collisions Ballistic pendulum: the height h can be found using conservation of mechanical energy after the object is embedded in the block.

Example: Ballistic Pendulum A projectile of mass m is fired with an initial speed v0 at the bob of a pendulum. The bob has mass M and is suspended by a rod of negligible mass. After the collision the projectile and bob stick together and swing at speed vf through an arc reaching height h. Find the height h.

Inelastic Collisions For collisions in two dimensions, conservation of momentum is applied separately along each axis:

Example: A Traffic Accident A car of mass m1 = 950 kg and a speed v1,i = 16 m/s approaches an intersection. A minivan of mass m2 = 1300 kg and speed v2,i = 21 m/s enters the same intersection. The cars collide and stick together. Find the direction q and final speed vf of the wrecked vehicles just after the collision.

Explosions An explosion in which the particles of a system move apart from each other after a brief, intense interaction, is the opposite of a collision. The explosive forces, which could be from an expanding spring or from expanding hot gases, are internal forces. If the system is isolated, its total momentum will be conserved during the explosion, so the net momentum of the fragments equals the initial momentum.

Elastic Collision 31.• A 722-kg car stopped at an intersection is rear-ended by a 1620-kg truck moving with a speed of 14.5 m/s. If the car was in neutral and its brakes were off, so that the collision is approximately elastic, find the final speed of both vehicles after the collision.

Solve 2 Equations in 2 Unknowns Solve the momentum equation for vc Substitute the expression for into the kinetic energy equation.

 Vt’ = 5.56 m/s Vt’ cannot be 14.5 m/s (WHY?) Substitute 5.56 m/s in for vt’ in the conservation of momentum equation to solve for vc’