# Imagine that a wall of your house is hit on one occasion by a boy on a skateboard and on another by a large juggernaut.

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1 A Resource for Free-standing Matheatics Qualifications Iagine that a wall of your house is hit on one occasion by a boy on a skateboard and on another by a large juggernaut. Suppose both had a speed of 1 s -1 at ipact. Which daages your house less? 1 s -1 Your answer to this question shows that when you odel a, you ust consider ore than just the velocities of the bodies involved. Their asses are also significant. Moentu When an object of ass kilogras oves with velocity v etres per second, its oentu is v (easured in kg s -1 or Ns). Moentu, like velocity, is a vector having both agnitude and diension. Whenever two objects are in contact, they exert equal and opposite forces on each other. In the case of a in which both objects can ove, the contact produces equal and opposite changes in oentu. Provided that no external force acts on either object, the total oentu of the two objects reains constant. For the Dynaics FSMQ, you will only need to use this property for objects oving in one diension. The Principle of Conservation of Linear Moentu When a occurs between two objects, the total oentu after is equal to the total oentu before the (assuing that there are no external forces). 1 v v 2 = 1 u u 2 u 1 s -1 u 2 s v 1 s -1 v 2 s -1 In soe s the two colliding objects separate after the, whereas in others they join together. In the first case the objects have different velocities after the whereas in the second case they ove together with the sae velocity after the ipact. When solving a proble involving a, draw a clear diagra showing the velocities before and after the ipact. In cases where the objects ove in opposite directions, take care to define one direction as the 'positive' direction an object oving in the opposite direction will have negative velocity. Soe exaples in real contexts follow. The Nuffield Foundation 1

2 A Resource for Free-standing Matheatics Qualifications Bowls In a gae of bowls a wood of ass hits the jack, of ass, which is stationary. The wood has a speed of 1.4 s -1 before the and a speed of 1 s -1 in the sae direction after the. What is the speed of the jack after the? 1.4 s -1 0 s -1 1 s -1 v s -1 By the principle of conservation of linear oentu v = v = = 0.4 v = 2 The jack has speed 2 s -1 after the. Skiers A skier of ass 75 kg who is oving at 8 s -1 collides with a skier of ass 60 kg oving at 4 s -1 in the sae direction. The faster skier's speed is reduced to 6 s -1 after the. What happens to the speed of the other skier? By the principle of conservation of linear oentu v = v = = 390 v = 6.5 The skier's speed is increased to 6.5 s -1 8 s kg 6 s -1 4 s -1 v s kg Tank A tank of ass 27.5 tonnes fires a shell of ass 7.5 kg with a speed of 300 s -1. Find the initial speed of recoil of the tank. By the principle of conservation of linear oentu v = v = 2250 v = The tank recoils with an initial speed of 8.18 cs -1 Toy trucks Two identical trucks, each of ass 200 g collide on a track. the they are oving towards each other as shown. the they ove together along the track at a speed of v s -1. Find v. 3 s -1 7 s kg 7.5 kg 0 s -1 0 s -1 v s s -1 Positive direction By the principle of conservation of linear oentu 0.4v = = 0.8 v = 2 The trucks ove at 2 s -1 in the direction of the faster truck. 200 g 200 g 200 g 200 g v s -1 The Nuffield Foundation 2

3 A Resource for Free-standing Matheatics Qualifications Worksheet 1 A boy and his toboggan have a cobined ass of 65 kg. They slide at a speed of 15 s -1 along a horizontal snow surface and then collide into a girl on her toboggan travelling in the sae direction at a speed of 7 s -1. The girl and her toboggan have a total ass of 50 kg., the speed of the boy and his toboggan is reduced to 9 s -1. The situations before and after the are shown below. Find the new speed, v s -1, of the girl and her toboggan after the. 15 s -1 7 s -1 9 s -1 v s kg 50 kg 65 kg 50 kg 2 A student experients with soe sall trolleys which can ove freely along a plastic track. a) The first which she investigates is arranged so that a trolley oves with initial speed 1 s -1 and collides with another which is stationary, as shown below. Both trolleys are identical and have ass kg. The trolleys stick together at and ove away together with speed v s -1. Use the principle of conservation of oentu to find v. u = 1 s -1 0 s -1 v s -1 b) In the second experient she investigates how the final speed, v s -1, of the cobined trolleys after the varies with the initial speed u s -1 of the oving trolley before ipact as shown below. Find an expression for v in ters of u. u s -1 0 s -1 v s -1 The Nuffield Foundation 3

4 A Resource for Free-standing Matheatics Qualifications 3 In a gae of bowls a wood of ass hits the jack, of ass, which is stationary. The wood has a speed of 1 s -1 before the and a speed of u s -1 after the. The jack has a speed of v s -1 after the as shown below. 1 s -1 u s -1 0 s -1 v s -1 a) Find an expression for v in ters of u. b) Draw a graph showing how v varies with u. Explain what would happen to the wood and jack at each of the points where your graph cuts the axes. 4 A an stands up in a sall rowing boat. He steps out of the rowing boat onto the river bank. he leaves the boat both the an and boat are stationary. The an has a ass 80 kg and the boat has a ass of 250 kg. When the an leaves the boat he has a speed of 1 s -1. The boat travels in the opposite direction with a speed of v s -1 as shown. Find v. 1 s -1 v s -1 5 A bullet is fired fro a gun. The ass of the gun is 2.5 kg; that of the bullet is 50 g. The bullet leaves the gun with speed 500 s -1. Find the initial speed of recoil of the gun. v s s kg 50 g 6 A girl, of ass 60 kg, jups onto a stationary skateboard of ass 3 kg. Just before she hits the skateboard, which is on horizontal ground, the girl is travelling in a horizontal direction with speed 4 s -1. Find the speed of the girl on the skateboard just after the. 7 Two snooker balls, both of ass 150 g collide directly. Just before the they are travelling with the speeds shown in the diagra. the the white snooker ball rebounds with a speed of 1.2 s -1. What happens to the black ball? 2 s -1 4 s -1 v s s -1 8 A 10 tonne train engine, which is rolling at a speed of 0.5 s -1, collides with a 2 tonne truck which is rolling in the opposite direction with a speed of 0.1 s -1. the the train and carriage ove together. Find the speed and the direction in which they ove after the. The Nuffield Foundation 4

5 A Resource for Free-standing Matheatics Qualifications Teacher Notes Unit Advanced Level, Dynaics Notes on Activity Pages 1 and 2 introduce the principle of conservation of oentu and show how this can be used to solve probles involving s. The worksheets on pages 3 and 4 give practice in solving siilar probles. Soe of these originally appeared in the book Mechanics 1 which was funded by the Nuffield Foundation and published by Longan in 1994 (ISBN X). Answers s -1 2 a) v = 1 2 s -1 b) v = 1 2 u 3 a) v = ( 1 u) v 5 b) Point (0, 5) 5 The wood is reduced to rest and the jack oves off with velocity 5 s s s s -1 (to 3 sf) 0 1 u Point (1, 0) The jack is at rest and the wood oves at 1 s -1 (the initial situation) 7 Black ball rebounds with velocity 3.2 s s -1 in the initial direction of the engine's otion. The Nuffield Foundation 5

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