# Linear Momentum Vectors again.

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Linear Momentum Vectors again

Review Equations for Motion Along One Dimension

Review Motion Equations for Constant Acceleration 1. 2. 3. 4.

Review 3 Laws of Motion If in Equilibrium If not in equilibrium
Change in Motion is Due to Force Force causes a change in acceleration

Review Work Energy

Review Law of conservation of energy Power efficiency

Collisions If an 18 wheeler hits a car, what direction will the wreckage move? What is the force between the 18 wheeler and the car?

Forces

Momentum Newtons 2nd law Linear momentum SI
Newton defined it as quantity of motion

Impulse When an object collides with another, the forces on the object will momentarily spike before returning back to zero.

Impulse

Impulse We now define impulse, J, as the change in momentum of a particle during a time interval SI unit

Example A ball with a mass of 0.40 kg is thrown against a brick wall. It hits the wall moving horizontally to the left at 30 m/s and rebounds horizontally to the right at 20 m/s. (a) find the impulse of the net force on the ball during the collision with the wall. (b) If the ball is in contact with the wall for 0.010s, find the average horizontal force that the wall exerts on the ball during impact.

Example

Conservation of Momentum
If a particle A hits particle B

Conservation of Momentum
If there are no external forces acting on the system

Conservation of Momentum
Change in momentum over time is zero The sum of momentums is constant

Conservation of Momentum
If there are no external forces acting on a system, Total Momentum of a system conserved

Example - Recoil A marksman holds a rifle of mass 3.00 kg loosely such that it’ll recoil freely. He fires a bullet of mass 5.00g horizontally with velocity relative to the ground of 300 m/s. What is the recoil velocity of the rifle?

Example - Recoil

Example – 2D example Two battling robots are on a frictionless surface. Robot A with mass 20 kg moves at 2.0 m/s parallel to the x axis. It collides with robot B, which has a mass of 12 kg. After the collision, robot A is moving at 1.0 m/s in a direction that makes an angle α=30o. What is the final velocity of robot B?

Example

Example

Example

Conservation of Momentum and Energy
Elastic Collisions – Collisions where the kinetic energies are conserved. When the particles are in contact, the energy is momentarily converted to elastic potential energy.

Conservation of Momentum and Energy
Inelastic Collisions – collisions where total kinetic energy after the collision is less than before the collision. Completely Inelastic Collisions- When the two particles stick together after a collision. Collisions can be partly inellastic

Completely Inelastic Collisions
Collisions where two objects will impact each other, but the objects stick together and move as one after the collision.

Completely Inelastic Collisions
Momentum is still conserved Find v in terms of v0

Completely Inelastic Collisions
Assume Particle B is initially at rest

Completely Inelastic Collisions
Kinetic Energy If B is at rest

Examples – Young and Freedman 8.37
At the intersection, a yellow subcompact car with mass travelling 950 kg east collides with a red pick up truck with mass 1900 kg travelling north. The two vehicles stick together and the wreckage travels 16.0 m/s 24o E of N. Calculate the speed of each of the vehicles. Assume frictionless.

Young and Freedman 8.37

Young and Freedman 8.37

Problem – Ballistic Pendulum
The ballistic pendulum is an apparatus to measure the speed of a fast moving projectile, such as a bullet. A bullet of mass 12g with velocity 380 m/s is fired into a large wooden block of mass 6.0 kg suspended by a chord of 70cm. (a) Find the height the block rises (b) the initial kinetic energy of the bullet (c) The kinetic energy of the bullet and block.

Problem – Ballistic Pendulum
Velocity after impact Kinetic energy after impact

Problem – Ballistic Pendulum
Kinetic energy after impact Converted to potential at highest point

Problem – Ballistic Pendulum

Elastic Collisions Momentum and Energy are conserved
Find v in terms of v0

Elastic Collisions – One Dimension
If particle B is at rest

Elastic Collisions – One Dimension
If particle B is at rest

Elastic Collisions – One Dimension
If particle B is at rest Substitute back

Elastic Collisions – One Dimension
If particle B is at rest

Elastic Collisions – One Dimension
If ma <<< mb really small

Elastic Collisions – One Dimension
If ma>>>mb If ma=mb

Example In a game of billiards a player wishes to sink a target ball in the cornet pocket. If the angle to the corner pocket is 35o, at what angle is the cue ball deflected? (Assume frictionless)

Example Mass is the same

Example

Problem – Serway 9-36 Two particles with masses m and 3m are moving towards each other along the x axis with the same initial speeds. Particle m is travelling towards the left and particle 3m is travelling towards the right. They undergo an elastic glancing collision such that particle m is moving downward after the collision at right angles from initial direction. (a) Find the final speeds of the two particles. (b) What is the angle θ at which particle 3m is scattered.

Elastic Collisions and relative velocity – One Dimensional

Elastic Collisions and relative velocity
In an elastic Collision, the relative velocities of the two objects have the same magnitude

Young and Freedman 8.42 A kg glider (puck on an air hockey table) is moving to the right with a speed of m/s. It has a head-on collision with a kg glider that is moving to the left with velocity 2.20 m/s. Find the final velocities of the two gliders. Assume elastic collision.

Problems- Young and Freedman 8.12
A bat strikes a 0.145kg baseball. Just before impact the ball is travelling horizontally to the right at 50.0 m/s and it leaves the bat travelling to the left at an angle of 30o above the horizontal with a speed of 65.0 m/s. Find the horizontal and vertical components of the average force on the ball if the ball and bat were in contact for 1.75 ms.

Giancoli 7-12 A 23 g bullet travelling at 230 m/s penetrates a 2.0kg block of wood and emerges cleanly at 170 m/s. If the wood is initially stationary on a frictionless surface, how fast does it move after the bullet emerges?

Serway 9.28 A 90.0 kg full back running east with a speed 5.0 m/s is tackled by a 95.0kg opponent running north at 3.00 m/s. If the collision is completely inelastic, (a) find the velocity of the players just after the tackle. (b) find the mechanical energy lost during the collision.

Prepare for pain Giancoli 7-78
A 0.25kg skeet (clay target) is fired at an angle of 30o to the horizon with a speed of 25 m/s. When it reaches its maximum height, it is hit from below by a 15g pellet traveling vertically upwards at 200 m/s. The pellet is embedded into the skeet. (a) how much higher does the skeet go up? (b) how much further does the skeet travel?

Assumptions so far Objects approximated to be point particles
Objects only undergo translational motion

Center of Mass Real objects also undergo rotational motion while undergoing translational motion. But there is one point which will move as if subjected to the same net force. We can treat the object as if all its mass was concentrated on a single point

Center of Mass Set an arbitrary origin point
Center of mass is the mass weighted average of the particles

Example A simplified water molecule is shown. The separation between the H and O atoms is d=9.57 x10-11m. Each hydrogen atom has a mass of 1.0 u and the oxygen atom has a mass of u. Find the position of the center of mass.

Example For ease set origin to one of the particles

Example

Center of Mass 1) if there is an axis of symmetry, the center of mass will lie along the axis. 2) the center of mass can be outside of the body

Center of Gravity The point of an object which gravity can be thought to act. This is conceptually different from center of mass For now the center of gravity of an object is also it’s center of mass.

Motion of Center of Mass

Motion of Center of Mass

External Forces and Center of Mass

Rockets Center of mass computations useful for when mass of a system changes with time

Orbits

Example James and Ramon are standing 20.0 m apart on a frozen pond. Ramon has a mass of kg and James has mass of 90.0 kg. Midway between the two is a mug of their favourite beverage. They pull on the ends of a light rope. When James has moved 6.0 m how far has Ramon moved?

No external forces! Center of Mass will not move!

Center of Mass will not move!
James moved 6m to the right

Problem – Young and Freedman 8.50
A 1200 kg station wagon is moving along a straight highway at 12.0 m/s. Another car with mass 1800kg and speed 20.0 m/s has its center of mass 40.0 m away. (a) Find the position of the center of mass of the two cars. (b) Find magnitude of total momentum of the system. (c) Find the speed of the center of mass of the system. (d) Find total momentum using center of mass.