Reading Quiz - Momentum 1. Which is true? Conservation of the total momentum of a system ___ 1. holds only when mechanical energy is conserved. ___ 2. holds for any system. ___ 3. follows from Newton’s second law. ___ 4. is equivalent to Newton’s third law.

2. A rocket is propelled forward by ejecting gas at high speed 2. A rocket is propelled forward by ejecting gas at high speed. The forward motion is a consequence of ___ 1. conservation of energy. ___ 2. conservation of momentum. ___ 3. both of the above. ___ 4. neither of the above.

3. The impulse delivered to a body by a force is ___ 1. defined only for interactions of short duration. ___ 2. equal to the change in momentum of the body. ___ 3. equal to the area under an F vs. x graph. ___ 4. defined only for elastic collisions.

4. In an elastic collision ___ 1. energy is conserved. ___ 2. momentum is conserved. ___ 3. the magnitude of the relative velocity is conserved. ___ 4. all of the above

Linear Momentum The linear momentum p of an object of mass m moving with velocity v is defined as: Note vector nature! Newton’s 2nd law can be re-expressed as:

Impulse Many forces are variable and act for a short period of time (as in collisions). A useful quantity is the impulse I of such a force: Equivalent average force of the impulsive force:

Conceptual Questions 1) Momentum is most closely related to ____ a) kinetic energy ____ b) potential energy ____ c) impulse ____ d) power ____ e) none of the above

2) An object that has momentum must also have ____ a) acceleration ____ b) impulse ____ c) kinetic energy ____ d) potential energy

3) Two equal-mass bullets traveling with the same speed strike a target. One of the bullets is rubber and bounces off, the other is metal and penetrates, coming to rest in the target. Which exerts the greater impulse on the target? ____ a) the rubber bullet ____ b) the metal bullet ____ c) both exert the same ____ d) not enough information ____ e) none of these

Quantitative Questions 1) What effect on its momentum does doubling the kinetic energy of a moving object have? 2) The head of a golf club is in contact with a 46 gram golf ball for 0.50 milliseconds, and as a result, the ball flies off at 70 m/s. Find the average force that was acting on the ball during the impact.

Conservation of Linear Momentum The total momentum of a system composed of many particles is simply the vector sum of the individual momentum of each particle. An isolated system is one in which the only forces present are those between the objects of the system. It follows from Newton’s 3rd law that the total momentum of an isolated system of bodies remains constant.

Quantitative Problems 1) A 13 gram bullet traveling 230 m/s penetrates a 2.0 kg block of wood and emerges going 170 m/s. If the block is stationary on a frictionless surface when hit, how fast does it move after the bullet emerges? 2) A spacecraft moving at 10 km/s breaks apart into 2 pieces of equal mass, one of which moves off at 4 km/s in a direction opposite to the original direction. Find the speed and direction of the other piece.

3) An astronaut outside an orbiting space craft uses a pistol that ejects a gas in order to maneuver in space. Suppose the astronaut in her space suit have a total mass of 100 kg and the pistol ejects 12 gm of gas per second at a speed of 650 m/s. How long should the astronaut operate the pistol in order to have a speed of 1 m/s?

Collisions Important: Momentum is always conserved in all collisions! Not energy or kinetic energy!! Elastic collision - one where total kinetic energy is conserved. Inelastic collision - one where total kinetic energy is not conserved. Completely inelastic collision - one in which the colliding bodies stick together after the collision.

Conceptual Question 1) In an elastic collision ____ a) momentum is conserved but not KE ____ b) KE is conserved but not momentum ____ c) momentum and KE are both conserved ____ d) neither momentum nor KE is conserved

Quantitative Problems 1) A pair of bumper cars collide elastically as one approaches the other directly from the rear. One has a mass of 450 kg and the other 550 kg. If the lighter one approaches at 4.5 m/s and other is moving at 3.7 m/s, calculate (a) their velocities after the collision, and (b) the change in momentum of each.

2) A 30 kg girl who is running at 3 m/s jumps on a stationary 10 kg sled on a frozen lake. How fast does the sled then move? 3) Two people, one of mass 75 kg and the other of mass 60 kg, sit in a rowboat of mass 80 kg. With the boat initially at rest, the two people, who have been sitting at opposite ends of the boat 2.0 m apart from each other, now exchange seats. How far and in what direction will the boat move? (Hint: it can be shown that the net force acting on a system of particles equals the total mass times the acceleration of the center of mass: )

Collisions in Higher Dimensions When a collision between 2 objects is not head on (called a glancing collision), the collision becomes 2- or 3-dimensional. Since momentum is a vector quantity, for these glancing collisions, each component of momentum must be individually conserved:

If collisions are also elastic, then the total kinetic energy is also conserved:

Conceptual Problem Two identical balls moving with the same speed towards each other along the x-axis suffer a glancing collision. After the collision, ____ a) they bounce back and move along the x- axis. ____ b) they must necessarily stick together and stop moving. ____ c) they can move off in any direction but must have equal and opposite velocities. ____ d) not enough information is given.

Quantitative Problems 1) Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in a perfectly elastic glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37 with its initial direction of motion and the velocity of the yellow disk is perpendicular to that of the orange disk (after the collision. Determine the final speed of each disk.

2) After a completely inelastic collision, two objects of the same mass and same initial speed are found to move away together at half their initial speed. Find the angle between the initial velocities of the objects.