Momentum

The force that is required to move an object or stop an object moving depends on: The object’s mass The object’s velocity Momentum is defined as the product of mass and velocity of an object. Is momentum a scalar quantity? No it is a vector. Since velocity has direction, momentum too will have a direction.

Change in Momentum ∆(mv) = Change in momentum ∆(mv) = final - initial = 0 - mv = - mv Ignoring negative sign: ∆(mv) = mv

Impulse Impulse is defined as a force acting on an object in a period of time. According to Newton’s second law of motion: Impulse = change in momentum

Example: Wall exerts a force of 10,000 N. The contact time is 0.01 s. Impulse = F t = 100 Ns

Momentum Change = Impulse (mv) = Ft ∆ mv = Ft = ∆p

Impulses and Contact Time

Rolling with the Punches Spreading impulse out over a longer time means that the force will be less; either way, the change in momentum of the boxing glove, fist, and arm will be the same

Questions: Q1: Superman needs to stop a train which is moving at 200 km/hr and has a mass of 10 tonnes. i) what is the change in momentum of the train? ii) Superman eventually stops the train in 5 sec. How much force did he need to exert on the train? Q2: A 1200kg car collides with a concrete wall at a speed of 15m/s and takes 0.06 sec to come to rest. a) What is change in momentum of car b) what is impulse of car c) what is force exerted on the car by the wall d) what would the force have been had the car bounced back at 3 m/s after being in contact for 0.06 sec?

Conservation of momentum In a collision the total momentum of the objects before the collision is equal to the total momentum after the collision provided that there are no external forces. Using Newton’s third law: Or:

Conservation of momentum Momentum Before = 0 Momentum After = 0 After firing, the opposite momenta cancel.

Conservation of Momentum - example Momenta are equal but opposite. M v = m V 4 v = x 300 = 3 v = 3 / 4 = 0.75 m / s

Definition - Collision Physicists employ the term collision in a broader sense than ordinary usage, applying it to any situation where objects interact for a certain period of time. A bat hitting a baseball, a radioactively emitted particle damaging DNA, and a gun and a bullet going their separate ways are all examples of collisions in this sense. Physical contact is not even required. A comet swinging past the sun on a hyperbolic orbit is considered to undergo a collision, even though it never touches the sun. All that matters is that the comet and the sun exerted gravitational forces on each other. The reason for broadening the term collision in this way is that all of these situations can be attacked mathematically using the same conservation laws in similar ways.

Collisions in Space This Hubble Space Telescope photo shows a small galaxy (yellow blob in the lower right) that has collided with a larger galaxy (spiral near the centre), producing a wave of star formation (blue track) due to the shock waves passing through the galaxies' clouds of gas. This is considered a collision in the physics sense, even though it is statistically certain that no star in either galaxy ever struck a star in the other. (This is because the stars are very small compared to the distances between them.)

Types of Collisions Elastic: when two bodies collide, they bounce back at the same speed that they collided. Partially elastic: when two bodies collide, they bounce back with a decreased velocity Inelastic: when two bodies collide they stick together

Elastic Collisions In elastic collisions no permanent deformation occurs; objects elastically rebound from each other In head-on elastic collisions between equal masses, velocities are exchanged.

Elastic Collisions on the Billiard Table

Collisions and Newton’s Cradle

Inelastic Collision Inelastic collisions-as between the arm and wooden plates are characterized by permanent deformation. The kinetic energy of the swinging arm is converted into heat and chemical energy (to break the bonds between atoms in the wood or possibly those in the hand).

Classic Inelastic Collision

Ballistic Pendulum A 10 gram bullet is fired from a rifle at a speed of 700 m/sec into a 1.50 kg block of a ballistic pendulum suspended by a string 4 meters long. After the collision, through what vertical distance does the block rise?

During the collision we can only use conservation of momentum. mbullet (vo bullet) + Mblock (vo block) = (m + M) vc(0.01)(700) + (1.5)(0) = ( ) vc7 = (1.51) vcvc = 4.64 m/sec

External forces Consider two balls colliding as shown Before collision After collision F F The only forces arising in this collision are forces acting on each ball by the other. Thus we say that there are no external forces acting, and conservation of momentum holds

Explosions Think about two trolleys of different masses exploding apart. From Newton’s third law it is clear that the trolleys are acted on by equal forces, in opposite directions. Both forces must act for the same time – the time the trolleys are in contact. These forces produce accelerations in inverse proportion to the masses, from Newton’s second law. So the bigger trolley has a smaller acceleration than the smaller one.

x components y components BEFORE COLLISION 2(-3) = -6 kg m/sec003(2)= 6kgm/sec AFTER COLLISION- 5 vc cos q5 vc sin qANALYSISconservation of momentum = - 5 vc cos q0 + 6 = 5 vc sin qvc cos q = 6/5vc sin q = 6/5

Elastic example Two identical balls (2kg) are moving toward each other at the same speed 3m/s. After collision, one of the balls is bounced backwards at 3m/s. Find velocity of other ball.

Partially elastic example Two identical balls (2kg) are moving toward each other at the same speed 3m/s. After collision, one of the balls is bounced backwards at 1m/s. Find velocity of other ball.

Inelastic example Two identical balls (2kg) are moving toward each other at the same speed 3m/s. After collision, the two balls stick together. Find the velocity of the cohesive mass.

Explosions In an explosion, an internal impulse acts in order to propel the parts of a system (often a single object) into a variety of directions. After the explosion, the individual parts of the system (which is often a collection of fragments from the original object) have momentum. If the vector sum of all individual parts of the system could be added together to determine the total momentum after the explosion, then it should be the same as the total momentum before the explosion. Just like in collisions, total system momentum is conserved in explosions as well.

Explosion example Two identical balls (2kg) stick together and are at rest initially. Some explosive is placed between them. After the explosion one ball moves forward with velocity of 4m/s. Find the velocity of the other ball.