Episode 221: Elastic collisions

Transcription

1 Episode 1: Elastic collisions This episode extends the idea of conservation of momentum to elastic collisions, in which, ecause KE is conserved, useful information can also e found y calculating the changes in KE of the colliding ojects. Summary Demonstration + Discussion: To introduce totally elastic collisions. (0 minutes) Student experiment: To test conservation of momentum and kinetic energy in an elastic collision. (0 minutes) Worked examples + student questions: Calculations of final velocities in elastic collisions and loss of KE in inelastic collisions. (0 minutes): Discussion: More astract prolems and situations which commonly cause difficulties. (15 minutes) Demonstrations: Showing some applications such as catching a all or finding the speed of an air rifle pellet. (15 minutes) Demonstration + Discussion: To introduce totally elastic collisions This follows a similar approach to the demonstration in Episode 0. TAP 0-1: Oserving collisions Demonstrate springy (elastic) collisions using trolleys, one of which has its spring-load released so its spring can soften the collisions (Alternatively, use air-track gliders with repelling magnets attached.) Direct a single trolley at a second, stationary trolley. The first trolley stops, the second moves off at the speed of the first. Momentum is conserved. Now try a light trolley colliding with a heavy one, and vice versa. What pattern is seen? A light trolley ounces ack from a heavier one (its momentum is negative); a heavier one moves on, ut at a slower speed. Students may accept that momentum is conserved; alternatively, with suitale light gates you should e ale to measure the initial speed of the one trolley and the final speeds of oth. Momentum conservation can e shown. (Or try using a computer and motion sensor.) Now ask: How do the trolleys know at what speed they must move? There are many cominations of velocity which conserve momentum; there must e something else going on here. Introduce the idea that kinetic energy (KE) is also involved, and that, in a springy collision, there is as much KE after as efore; in other words, KE is conserved. Ask whether KE is conserved in an explosion (oviously not; it is created in the explosion in the change from chemical PE to KE) and in an inelastic collision (the total amount decreases, ut you may need to work through a numerical example to show this). 1

2 Discuss where KE comes from in an explosion (from energy stored in a squashed spring, chemical explosive or whatever), and where it goes to in an inelastic collision (work is done in deforming material leads to heating; some sound). Terminology: Usually, elastic is taken to imply that KE is conserved. In some texts, this is written as perfectly elastic. Inelastic descries a collision in which some KE is lost. Students should learn to use these terms, rather than springy and sticky. Student experiment: To test conservation of momentum and kinetic energy in an elastic collision Ask students to carry out an experiment to determine whether, in a springy collision etween trolleys or gliders, momentum and KE are oth truly conserved. They can use the same approach as the experiment in Episode 0, ut they will have to calculate values of KE as well as momentum. Worked examples + student questions: Calculations of final velocities in elastic collisions and loss of KE in inelastic collisions Work through examples involving elastic collisions to show: Conservation of KE and momentum in an elastic collision, when all values of mass and velocity are known. Calculation of final velocities in an elastic collision. Note that the second type requires solution of simultaneous equations, one of which is a quadratic; students may find this difficult, so check that it is required y the specification that you are following. Non-conservation of KE in an inelastic collision. Students can now work through more examples. TAP 1-1: Worked examples in momentum: elastic and inelastic collisions TAP 1-: Student Questions Discussion: More astract prolems and situations which commonly cause difficulties Reinforce students ideas y discussing some of the following: How a rocket ship works, as a controlled explosion in which reaction mass travels ackwards. The rocket needs nothing to lift off except the expended fuel.

3 There is a video of this with a man sitting on a cart firing a fire extinguisher at (1N.10 [M-1 W] Fire Extinguisher Wagon ) The same site shows another video of two people on roller-lades throwing a cushion ack and forth. Discuss situations in which the Earth is involved, as it may appear that momentum is not conserved. Where does momentum come from or go to in these situations? It helps to think aout the forces involved. You push a car to start it moving. (Your feet push ack on the Earth, so that its momentum also changes, in the opposite direction. This is equivalent to an explosion.) When a all falls, it accelerates, i.e. it gains momentum. (The Earth is also accelerated minutely in the opposite direction, so momentum is conserved. The force is gravity.) When a all ounces off a wall, its momentum is reversed. (Momentum is transferred to the wall + Earth y the contact force.) When a all rolls to a halt, it loses momentum. (Its momentum is transferred to the Earth via friction). These all emphasise the need to think of the closed system with which we are concerned. Emphasise that momentum is always conserved, ut KE is not. One way to think of this is that KE is just one form of energy, so it can e transformed; there is only one form of momentum, so it cannot e transformed into anything else. Demonstrations: Showing some applications such as catching a all or finding the speed of an air rifle pellet TAP 1-3: Momentum demonstrations The following is a rief summary of the Resourceful Physics activities given in the link in TAP 1-3: RP13: Pendulum on a trolley to show conservation of momentum. This could e filmed with a wecam and the film studied in slow motion to great effect. There is a link here etween the motion of the trolley and the motion of a rowing oat, which shoots forwards when the rowers move ack on their slides, rather than when they pull on their oars. RP18: Momentum in catching compare to deforming arriers and elastic ones. This contrasts the momentum change on catching something and the greater change on reflecting it ack. RP 10: If you have access to an air rifle demonstration, this is a lovely way to round off the topic. RP 4: Speed of a cricket all can e used in place of the air rifle experiment 3

4 TAP 1-1: Worked examples in momentum: elastic and inelastic collisions 1. When an oject of mass M and velocity u collides head-on elastically with a stationary oject of mass m then mass m moves off with velocity v given y: - v M = u and the mass M changes speed from u to v a where v a is given y:- M m v a = u a) A golf clu head has a mass very much larger than that of the all. Use the aove formula to show the golf all will e driven from the tee at aout twice the speed of the golf clu head. ) An alpha particle makes a head on collision with a stationary helium atom. Show the alpha particle will stop and the helium atom will move off with the original speed of the alpha particle. (Assume an alpha particle and helium nucleus have the same mass) c) A fast moving electron collides with a stationary heavy atom. Show the electron ounces ack with nearly the same speed. A cricket all, mass 0.5 kg, was owled at 50 m s -1 at a atsman who misreads the all and the 5 kg at is knocked out of his hands, the all reounds at 5 m s -1. a) What is the initial momentum of the all? ) What is the momentum of the all just after it hit the at? c) How much momentum is transferred to the at? d) Calculate the velocity of the at as it leaves the atsman. e) Calculate the kinetic energies efore and after the collision of at and all. Is the collision elastic or inelastic? 3. A all of mass 0.0 kg is dropped from a height of 3. m onto a flat surface which it hits at 8.0 m s -1. It reounds to 1.8 m. (g = 9.8 m s - ) a) What is the reound speed just after impact? ) What is the change in energy of the all? c) What momentum change has the all etween just touching the surface and leaving it? 4. In Octoer 1999 the United Nations announced that the world s population had reached 4

5 6 illion (6 x 10 9 ) and that the population of the People s Repulic of China was rapidly approaching 1.3 illion. Imagine a situation in which the entire population of China were gathered together and, all at the same time, they each jumped as high as they were ale. Estimate the speed that would e (temporarily) imparted to the Earth as a result of this jump. Show all your calculations and reasoning and state clearly any assumptions you have made. Take the mass of the Earth to e 6 x 10 4 kg. 5

6 Answers and Worked Solutions 1. a) The all has a much smaller mass than the clu head so it can e ignored in M v = u M so v = u and v. So the speed is twice that of the clu head. As the all does M = u not have zero mass the all speed will e slightly under u M m ) The alpha particle speed after the collision is given y v a = u as M = m the speed M will e zero. The helium nucleus will move off with speed given y v = u and as M M = m then v = u so v u M =. Therefore the alpha particle stops and the helium nucleus moves off with the same speed. M m M c) v a = u m can e ignored so v a = u and v a = u M The larger mass will hardly move. a) Initial momentum of the all = 0.5 x 50 = 5 kg m s -1 ) Momentum of the all just after it hit the at = 0.5 x (-5) = -1.5 kg m s -1 (Taking positive velocity to the right.) c) Momentum transferred to the at =5 - (-1.5) = 37.5 kg m s -1 d) Momentum of at = 37.5 kg m s -1 so velocity of at = (37.5)/5 = 7.5 m s -1 e) Calculate the kinetic energies efore and after the collision of at and all. Is the collision elastic or inelastic? 1 1 KE = mv so efore collision KE = 0.5(50) and KE = 65 J 1 After the collision KE of all is KE = 0.5(5) so KE = J = 156 J to 3sf 1 After collision Bat KE = KE = 5(7.5) so KE = J = 141 J to 3sf Total KE after collision is =96.9 J = 97 J to 3sf so the collision is inelastic as KE efore > KE after the collision 3. a) This is an energy reminder v = gh so x(9.8) x(1.8) velocity = m s -1 v = and speed = 5.9 m s -1 so ) 1 KE = mv so efore impact KE = 0.5 x 0.0 x 8.0 x 8.0 = 6.4 J after impact KE = 0.5 x 0.0 x (-5.9) x (-5.9) = 3.5 J so KE change is =..9J 6

7 c) Momentum efore collision = 0. x 8.0 = 1.6 kg m s -1 Momentum of all after = 0. x (-5.9) = -1. Change in momentum = (-1.) =.8 kg m s This question shows the effect of many small masses on a very much larger mass Estimate average mass of person m = 50 kg (NB Population will contain many children) Jump height _h = 0. m Take-off speed v = gh so v = x10x0. = m s -1 (accept any reasonale method or lind guess of aout 1 to 3 m s -1 ) Total upward momentum of all jumpers: p =1.3 x 10 9 x 50 kg x m s -1 = 1.3 x kg m s -1 To conserve momentum, Earth must acquire equal momentum in the opposite direction. Earth, mass M, acquires speed v = p/m = 1 x 3 x / 6 x 10 4 = x m s -1 i.e. velocity of the Earth is negligile. An elastic or inelastic jump? External References Question 1 was an adaptation of question 55(L) from Revised Nuffield Advanced Physics Unit A Question 3 was an adaptation of question 50(I) from Revised Nuffield Advanced Physics Unit A Question 4 is taken from Salters Horners Advanced Physics, Section TRA, Additional sheet 8 & 9 7

8 TAP 1- : Student Questions Elastic Collision Questions 1. A trolley of mass 1 kg rolls along a level, frictionless ramp at a speed of 6 m s -1. It collides with a second trolley of mass kg which is initially at rest. The first trolley reounds at a speed of m s -1. a) Find, y conservation of momentum, the velocity of the second trolley after the collision. ) Compare the kinetic energy efore and after the collision. Is the collision elastic?. A owling all of mass 5 kg strikes a skittle of mass 1 kg. The owling all is travelling at 3 m s -1 efore the collision, which is elastic. Find the velocity of the all and the skittle after the collision. (NB in this question ignore the rolling motion of the all assume it is sliding). 8

9 Practical Advice These are quite advanced and may not e appropriate for all specifications. Answers and Worked Solutions 1. a) Let us use the initial direction of the first trolley as the positive direction. Then momentum efore = m 1 u 1 = 1 6 = 6 kg m s -1. Afterwards let the velocities e v 1 and v and the masses are still m 1 and m. Then: 6 = m 1 v 1 +m v 6 = 1 x (-) + x v v = 4 m s -1 (Note the care with the signs) ) KE efore = ½m 1 u = 1/ = 18 J KE after = ½m 1 v 1 + ½m v = ½ 1 (-) + ½ (4) = + 16 = 18 J So KE is conserved and the collision is elastic.. Conservation of momentum: m u = m v + m s v s where the suscripts s and refer to the skittle and the all. This gives: 15 kg m s -1 = 5v + v s Conservation of KE: ½m u = ½ m v + ½ m s v s ½ 5 (3) = ½ 5 v + ½ 1 v s.5 J =.5v + 0.5v s To solve this we use the first equation to eliminate v s, v s = (15-5v ), and sustitute in the second equation:.5 =.5 v (15-5v ) We ll multiply through y 0.4 to keep the numers easy: 9 = v + 5(3 - v ie ) 9 = v v - 30v 6v - 30v + 36 = 0 v - 5v + 6 = 0 (v - 3)(v -) = 0 v = m s -1 (or 3 m s -1 ut that was the original value) Sustituting ack in the first equation, v s = 1 m s -1 9

10 TAP 1-3: Momentum demonstrations 13. Pendulum on a trolley - momentum conservation Hang a pendulum on a dynamics trolley or on the rider of a linear air track and oserve its motion as the trolley moves along. Apparatus required: Linear air track Pendulum supported on an air-track rider 18. Momentum in catching This simple demonstration emphasises the vector nature of momentum. Use a all (soccer or netall) and throw it to a pupil. Get them first to catch it and throw it ack and then to punch it ackwards without catching it first. Ask students which require the iggest force - catching or punching it ack. It should e clear that it is second and that this gives the iggest change of momentum. This demonstration hopefully emphasises the velocity change from u to -u and so a greater change of momentum than simply stopping the all. Safety Consider whether or not this activity can e done safely in your la with your pupils. Theory: Momentum change = mu - (-mu) = mu when it is punched ack [NB this fact will e used when calculating the pressure due to a gas using the kinetic theory of gases.] and momentum change = mu when it is caught Apparatus required: Netall or footall 10. The air rifle and momentum How safe are these two experiments? The writer elieves that if done with care and due regard to everyone s safety they are fine! So here they are. (a) Fix two timing gates - simply a plastic frame with a strip of aluminium foil across the centre - a metre apart and connected to a scalar. Fire an air rifle pellet through the two gates so that it starts the timer when the first strip of aluminium foil is roken and stops it when the second is roken. The time for the pellet to travel the one metre etween the two pieces of foil is given directly and the speed of the ullet can easily e worked out Safety - the rifle should e clamped to a aseoard and a means of catching the pellet should e found - ox of polystyrene acked with a wooden oard is suitale for this. Use a safety screen. 10

11 () Using the same mounted air rifle fire a pellet into a lock of Plasticine mounted on a model railway truck on rails. The runway should e tilted to compensate for friction and the speed of the truck (V) after impact should e found y timing its first 0 cm of movement.! Wear safety spectacles This experiment must not e performed unless the air rifle is securely olted to the aseoard of the track. Use a safety screen. All persons present must e positioned ehind the muzzle of the rifle. Rememer that trolley runways require two people to carry and manipulate them. Theory: Knowing the mass of a pellet (m) and the mass of the truck and Plasticine (M) the speed of the pellet can e found (mv + MV = 0) Apparatus required: Mounted air rifle with timing gates Truck loaded with Plasticine on rack on trolley runway Scaler Ruler Stopwatch Backstop External References This activity is taken from Resourceful Physics 11