Momentum and Collisions 1

Transcription

1 Momentum and Collisions 1 In everyday life Collisions and other transfers of momentum occur frequently o Sports tennis, racquetball, football o Traffic accidents What to expect You will analyze momentum and collisions between 2+ objects You will consider the mass and velocity of one or more objects and the conservation of momentum and energy Physics vs. the rest of the world! Momentum used in everyday speed is similar to the physics meaning. A bicycle coasting down a hill accelerates o picking up speed gaining momentum The faster you move, the more momentum you have and the more difficult it is to stop. Comparing momentum A bowling ball being rolled down a lane at the bowling alley A playground ball being rolled down another lane at the same speed Which has the greater momentum? Why? Can a smaller object ever have greater momentum? If so, when? Linear momentum Consider a soccer game where a player heads a moving ball and the ball s velocity changes rapidly. 1-dimensional motion equations predict the motion of the ball before and after it is struck Force and Newton s laws can be used to calculate how the motion of the ball changes Linear momentum of an object with mass (m) moving with a velocity (v) is defined as mass x velocity. p = mv momentum = mass x velocity p? Newton called momentum quantity of motion. German mathematician Gottfried Leibniz used the term progress to mean the quantity of motion with which a body proceeds in a certain direction. Momentum is a vector quantity! Its direction matches that of the velocity. Momentum has dimensions of mass x length/time kg m/s Sample Problem A A 2250 kg pickup truck has a velocity of 25 m/s to the east. What is the momentum of the truck?

2 Momentum and Collisions 2 A change in momentum takes force and time Momentum changes when a net force is applied. If momentum changes, forces are created. Quick changes mean large forces: It takes more force to stop a fast moving ball than a slower one. It takes more force to stop a real dump truck than a toy dump truck moving at the same velocity. Changes in momentum are closely related to force. Impulse-momentum Newton originally expressed his 2 nd law of motion as: Force = change in momentum time interval F = p t Rearrange this formula and you find the change in momentum in terms of the net external force and the time interval required for the change: p = F t Impulse-momentum theorem Impulse-momentum theorem = F t = p = mv f mv i A net external force, F, applied to an object for a certain time interval, t, will cause a change in the object s momentum equal to the product of the force and the time interval. F t is called the impulse of the force, F, for the time interval, t Why impulse is important Extending the time interval over which a constant force is applied allows a smaller force to cause a greater change in momentum than would result if the force were applied for a short period of time. Sample Problem B A 1400 kg car moving westward with a velocity of 15 m/s collides with a utility pole and is brought to rest in 0.30 s. Find the force exerted on the car during the collision. Baseball, softball, croquet, billiards, karate

3 Momentum and Collisions 3 Stopping times and distances Highway safety engineers use the impulsemomentum theorem to determine stopping distances and safe following distances for cars and trucks. o Loaded truck has twice the mass as an empty truck, so has twice the momentum. o Stopping time will also be doubled. Sample Problem C A 2240 kg car traveling to the west slows down uniformly from 20.0 m/s to 5.00 m/s. How long does it take the car to decelerate if the force on the car is 8410 N to the east? How far does the car travel during the deceleration? Longer time interval If the time interval over which an impact occurs is lengthened, the amount of force is decreased. A longer time interval requires a smaller force to achieve the same change in momentum. Consider poor Humpty Dumpty Humpty Dumpty met his unfortunate demise when he very unwisely jumped from the top of the wall to the garden floor below (as demonstrated to the right by his cousin Eggbert, who also unwisely jumped to a plate). Now, had Humpty and Eggbert followed in the example set by Eggland and chosen a nice comfy pillow to jump on, all would have been well.

4 Section 2: Conservation of Momentum Momentum and Collisions 4 Momentum of 2+ interacting objects Conservation of momentum The momentum of ball A plus the momentum of ball B before the collision equals the momentum of ball A plus the momentum of ball B after the collision. Moving billiard ball (A) hits a stationary ball (B) Before the collision, what was ball B s momentum? Zero (v = 0 m/s) During the collision, ball B gains momentum while ball A loses momentum. After collision, ball A is now stationary, but ball B is moving. What is ball A s momentum after the collision? Zero (v = 0 m/s) The momentum that ball A loses is exactly equal to the momentum that ball B gains. The momentum of each ball changes due to the collision, but the total momentum of the two balls together remains constant. What if there are 3 objects? In general, the total momentum remains constant for a system of objects that interact with each other. If a 3 rd object exerted a force on either ball A or ball B during the collision, the total momentum of ball A, ball B, and the 3 rd object would remain constant. o You must include all forces in an interaction in real world examples, including friction. This relationship is true for all interactions between isolated objects and is known as the law of conservation of momentum. m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f For an isolated system: The total momentum of all objects interacting with one another remains constant regardless of the nature of the forces between the objects. Total initial momentum = total final momentum Sample Problem D A 76 kg boater, initially at rest in a stationary 45 kg boat, steps out of the boat and onto the dock. If the boater moves out of the boat with a velocity of 2.5 m/s to the right, what is the final velocity of the boat? Momentum is conserved when 2 balls collide 2 objects push away from each other o You jumping up from the Earth o 2 stationary skaters facing each other, who then push away from each other

5 Momentum and Collisions 5 Newton s 3 rd law and conservation of momentum 2 isolated bumper cars before and after the crash The impulse-momentum theorem describes the change in momentum of one of the cars (at a time): F t = m 1 v 1,f m 1 v 1,i (and the same for car 2) F1 is the force of car 2 on car 1; F2 is the force of car 1 on car 2. Because the only forces acting in the collisions are the two bumper cars acting on each other, Newton s 3 rd law tells us that the force on car 1 is equal to and opposite the force on car 2. Forces take place in the same interval. Physics class versus reality Real life is different from class In class, collisions are treated as if the forces are constant Reality: the magnitudes of the forces change throughout the collision increasing, reaching a maximum, and then decreasing Back to momentum The impulse on car 1 is equal and opposite to the impulse on car 2 o True of all collisions or interactions between 2 isolated objects). Because impulse is equal to the change in momentum, the change in momentum in car 1 is equal and opposite to the change in momentum for car 2. If the momentum of one object increases after a collision, then the other object s momentum decreases by an equal amount. Section 3: Elastic and Inelastic Collisions Momentum change Compare these scenarios: A rubber ball is bounced on the gym floor and bounces upward A clay ball is bounced on the gym floor and hits with a thud Collisions 2 types: 2 objects collide and stick together so that they travel together after the impact 2 objects collide and move apart with 2 different velocities

6 Momentum and Collisions 6 Momentum after collision Total momentum remains constant in any type of collision Total kinetic energy is generally not conserved in a collision because some kinetic energy is converted to internal energy when the objects deform Perfectly inelastic collisions Perfectly inelastic two objects collide and move together as one mass o Arrow hitting target, meteorite colliding and imbedding in Earth s surface Momentum in inelastic collisions The objects become essentially one object after the collision Final mass is equal to the combined mass of the 2 objects Objects move with the same velocity after colliding Watch your signs to indicate direction when using this equation (right is generally positive, while left is negative) m 1 v 1i + m 2 v 2i = (m 1 + m 2 )v f Sample Problem E A 1850 kg luxury sedan stopped at a traffic light is struck from the rear by a compact car with a mass of 975 kg. The 2 cars become entangled as a result of the collision. If the compact car was moving at a velocity of 22.0 m/s to the north before the collision, what is the velocity of the entangled mass after the collision? Kinetic energy in inelastic collisions The total kinetic energy does not remain constant when the objects collide and stick together Energy is transformed to sound energy and internal energy as the objects deform during the collision

7 Momentum and Collisions 7 Physics oddities (aka weirdness!) Elastic In English, elastic refers to an object returning to or keeping its original shape In physics, an elastic material is one in which the work done to deform the material during a collision is equal to the work done to reform the material to its original shape Work done on an inelastic material is converted to other forms of energy, such as sound or heat. Kinetic energy in inelastic collisions The kinetic energy initially (KE i ) is found by solving for the KE of the objects individually and then adding them together. Kinetic energy final (KE f ) is found by adding the 2 masses together and multiplying by the final velocity of the combined mass, squared. KE f = ½ (m 1 + m 2 )v f 2 Sample Problem F Two clay balls collide head-on in a perfectly inelastic collision. The first ball has a mass of kg and an initial velocity of 4.00 m/s to the right. The second ball has a mass of kg and an initial velocity of 3.00 m/s to the left. What is the decrease in kinetic energy during the collision? Elastic Collisions Elastic collisions occur when the objects remain separate after the collision, like a soccer player kicking a soccer ball. The two objects collide and return to their original shapes with no change in total kinetic energy. In an elastic collision, both the total momentum and total kinetic energy are conserved. Realistic collisions Outside our make-believe world Most collisions fall in a category between perfectly inelastic collisions and elastic collisions o Inelastic collisions Inelastic collisions result in the decrease in total KE o All of our problems will either be perfectly inelastic (traveling as one) or elastic (no loss in KE) Most collisions are not perfectly inelastic o Colliding objects do not usually stick together and continue moving as one Most collisions are not elastic o no loss of kinetic energy Most collisions will result in some decrease of kinetic energy o Billiard balls colliding lose some kinetic energy o Football is deformed slightly when it is kicked o Some KE is converted to internal elastic PE o Some KE is converted to sound energy Any collision that produces sound is not elastic o Sound signifies a decrease in kinetic energy

8 Momentum and Collisions 8 Kinetic energy in elastic collisions Ball A is moving right; ball B is moving left at a greater speed Two balls collide; ball A now moves to the left and ball B now moves to the right Momentum of ball A is now greater than the momentum of ball B Sample Problem G A kg marble moving to the right at m/s makes an elastic headon collision with a kg shooter marble moving to the left at m/s. After the collision, the smaller marble moves to the left at m/s. Assume that neither marble rotates before or after the collision and that both marbles are moving on a frictionless surface. What is the velocity of the kg marble after the collision? Total momentum of the system is unchanged Total momentum is always constant throughout the collisions If the collision is perfectly elastic, KE after the collision is the same as before the collision m 1 v 1,i + m 2 v 2,i = m 1 v 1,f + m 2 v 2,f ½m 1 v 1,i 2 + ½ m 2 v 2,i 2 = ½m 1 v 1,f 2 + ½ m 2 v 2,f 2 Remember, v is positive if the object moves to the right and negative if it moves to the left Inelastic vs. elastic collisions Which ball has the greater force on the floor a rubber ball or a clay ball? Bounces and momentum Change in momentum is always greater when objects bounce compared to no bounce Assume both balls have a mass of 1 kg and both balls are moving at 2 m/s when they hit the floor. Rubber ball goes from 2 kg m/s to + 2 kg m/s, for a change of +4 kg m/s Clay ball goes from 2 kg m/s to 0 kg m/s, for a change of +2 kg m/s

9 Momentum and Collisions 9 The rest of the bounce and momentum story A change in momentum of +4 kg m/s only tells you the product of force and time, not force or time individually. The product of a force and the time the force acts is called the impulse. Rearrange the momentum form of Newton s 2 nd law (F= p/t) to get F t = p Impulse (F t) is measured in N s Change in momentum is still kg m/s Collision Cheat Sheet