E k = ½ m v 2. (J) (kg) (m s -1 ) FXA KINETIC ENERGY (E k ) 1. Candidates should be able to : This is the energy possessed by a moving object.

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1 KINETIC ENERGY (E k ) 1 Candidates should be able to : This is the energy possessed by a oing object. Select and apply the equation for kinetic energy : E k = ½ 2 KINETIC ENERGY = ½ x MASS x SPEED 2 E k = ½ 2 Apply the definition of work done to derie the equation for the change in graitational potential energy. (J) (kg) ( s -1 ) DERIVATION OF E k = ½ 2 Select and apply the equation for the change in graitational potential energy near the Earth s surface : u = 0 F a F ΔEp = g Δh s Analyse probles where there is an exchange between graitational potential energy and kinetic energy. Apply the principle of conseration of energy to deterine the speed of an object falling in the Earth s graitational field. Consider an object of ass () acted on by a constant force (F) which gies it a constant acceleration (a), and increases its elocity fro rest to a final alue () oer a distance (s). kinetic energy gained by object, E k = work done by force F E k = force x distance oed in the force direction E k = F x s E k = as (since F = a) But 2 = u 2 + 2as = 2as (since u = 0) So as = ½ 2 Therefore : E k = ½ 2

2 GRAVITATIONAL POTENTIAL ENERGY (E p ) 2 PRACTICE QUESTIONS (1) 1 A otorcycle has 5 x 10 4 J of kinetic energy. If the brakes delier a braking force of 650 N, calculate the shortest stopping distance for the otorcycle. When an object is lifted to a higher position aboe the ground, work is done against the force of graity and this transfers graitational potential energy to the object (Strictly speaking it is the Earthobject syste which gains graitational potential energy). 2 Calculate the increase in kinetic energy of a ehicle of ass 1200 kg when it accelerates fro 10 s -1 to 25 s -1. GRAVITATIONAL POTENTIAL ENERGY (E p ) is the energy possessed by an object due to its position or height aboe the Earth. 3 A bullet of ass 8.0 g is gien 160 J of kinetic energy when it is fired fro a gun. Calculate the elocity of the bullet as it leaes the gun barrel. 4 Use the internet to find approxiate asses and speeds of each of the following and hence estiate a alue for their kinetic energy : A loaded faily car traelling along a otorway at the speed liit. A ale Olypic 100 sprinter. A fully laden Jubo jet aircraft at noral cruising speed. Consider an object of ass () which is raised through height (h) aboe the ground. E p gained = work done in lifting by object = force x distance oed = object weight x height Lifted = g x h Therefore : DERIVATION OF E p = g h g h g g ground A tennis ball sered by a Wibledon chapion. Change in graitational potential energy (E p ) is gien by : An electron traelling at 6.8 x 10 8 s -1 as it exits a linear accelerator. E p = g h The Earth oing at its orbital speed around the Sun.

3 3 PRACTICE QUESTIONS (2) E p E p E k 1 An athlete of ass 76 kg runs up a hill in the Lake District. He starts at a point 200 aboe sea-leel and finishes at the suit which is 950 aboe sea-leel. Calculate the athlete s increase in graitational potential energy (g = 9.81 N kg -1 ). Work is done by the otor to pull the cart oer the first hill. E k Ek Ep 2 A catapult has 25 J of elastic energy. If all its elastic energy is used to project a arble of ass 5.0 g ertically upwards, what is the axiu height reached by the arble? (Take g = 9.81 N kg-1). The diagra aboe shows the cart at arious positions on the roller coaster as well as the energy transforations which occur at each point. 3 A ball of ass 0.25 kg drops fro a height of 12 and rebounds to a height of 8.5. Assuing negligible air resistance, calculate the energy lost on ipact with the ground (g = 9.81 N kg -1 ). E p -E k TRANSFORMATIONS ON A ROLLER COASTER At the top of the first hill, the cart is oentarily stationary and only has E p. As it accelerates down the slope it loses E p and gains E k (i.e. E p is being transfored into E k ). A roller coaster proides us with an excellent exaple of transforations of graitational potential energy to kinetic energy and ice ersa. A otor pulls the cart oer the top of the first hill. It then runs down the other side, accelerating as it goes. The second hill is lower than the first and the cart is oing just fast enough to ake it oer the top and once again accelerate down the second slope. The work done on the cart by the otor is transferred to graitational potential energy which is then transfored to kinetic energy as it speeds down the slope. At the botto of the slope, the cart s initial E p has been transfored into E k. As it runs up the second hill, work is done against graity and so the cart slows down. It loses E k and gains E p (i.e. E k is being transfored into E p ). As the cart oes along, soe of its kinetic energy is used to do work against friction and air resistance. Thus, soe of the cart s kinetic energy is transfored into heat and sound energy and it is therefore unaailable for transforation into graitational potential energy. For this reason, the cart cannot return to its original height and so the second hill ust be lower than the first and the third ust be lower than the second and so on.

4 4 There are any other exaples of Ep Ek transforation. In the Winter Olypics Ski Jup copetitors slide down a long snowcoered slope fro a great height. E p is transfored into E k and this is used by the juper to achiee the axiu possible jup distance. In the case of an oscillating siple pendulu, there is a continuous interchange of E p and E k as the bob oes fro C A B A C. E p(ax) = gh E k = 0 At C and B where the bob oentarily coes to rest, E k(ax) = gh the bob has zero Ek and E p = 0 axiu E p and at A where the bob is oing at its axiu elocity, E p is zero and E k = gh has its ax alue. E p(ax) = gh E K = 0 If an object is thrown upwards, its initial E k is transfored into E p as it rises and slows down. Eentually, when it reaches the axiu height, all the E k is transfored into E p (assuing zero air resistance). = 0 E p is ax E k is zero E k E p E k is ax E p is zero As we hae seen in all the exaples considered, when an object falls its E p decreases and its E k increases. Assuing no energy is lost in the process : E p lost = E k gained This relation can be used to sole a ariety of probles, such as the elocity attained by an object when it falls fro a gien height. Consider an object of ass () which Falls fro a height (h) aboe the ground. g h E k gained = E p lost ½ 2 = gh Fro which : = (2gh)

5 PRACTICE QUESTIONS (3) 1 A high dier reaches the highest point in his jup at which his centre of graity is 11.4 aboe the water surface. Assuing that all the dier s graitational potential energy is transfored into kinetic energy during the die, calculate the elocity with which he enters the water (Take g = 9.81 s -2 ). 2 An object of ass 0.75 kg is projected ertically upwards with a elocity of 12 s -1. If it reaches a height of 6.75, calculate the energy loss caused by air resistance (Take g = 9.81 s -2 ). HOMEWORK QUESTIONS 5 1 Describe the energy changes that occur in each of the following : (a) A cyclist freewheels fro rest down a hill and then uses the brakes to stop at the botto. (b) The bob on a siple pendulu is displaced fro equilibriu with the thread taut and then released. The bob swings across to axiu displaceent on the other side of the equilibriu position. 2 A rock falls fro the top of a 75 high cliff and strikes the ground at the botto with a elocity of 35 s -1. (a) What percentage of the rock s initial graitational potential energy is transfored into kinetic energy as a result of the fall? 3 A steel ball bearing of ass 0.05 kg at a height of 2.0 aboe a steel table is released fro rest and it is found to rebound to a height of 1.8. Calculate : (a) The graitational potential energy lost during the fall. (b) The kinetic energy and elocity of the ball bearing just before ipact. (c) The graitational potential energy gained by the ball bearing when it rebounds to a height of 1.8. (d) The ball bearing s rebound elocity. (Take g = 9.81 s -2 ). (b) Explain what happens to the rest of the rock s initial energy. 3 The diagra opposite shows the ertical section through a ski track. A skier of ass 76 kg starts fro rest at A. Assuing friction to be negligible, calculate : (a) The skier s elocity at point B. (b) The axiu horizontal distance (s) fro point O that the skier reaches.

6 6 4 A toy car of ass 0.50 kg is released fro point A on a frictionless track. Calculate the car s : (a) Kinetic energy at point B. (b) Velocity at point B. (c) Graitational potential energy at point C. (d) Kinetic energy at point C. (e) Velocity at point C. (Take g = 9.81 s -2 ). 5 A roller coaster cart of total ass 1500 kg oing with an initial elocity of 2 s -1 descends through a height of 70 to reach a elocity of 36 s -1 after traelling a distance of 120 along the track. (a) Calculate : (i) Its loss of graitational potential energy. (ii) Its gain of kinetic energy. (b) Show that the aerage frictional force acting on the cart during the descent was 500 N.