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Vectors<br />The length of the line represents:<br />the magnitude and,<br />The direction of the line segment represents:<br />the direction of the vector quantity.<br />
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Kinematic Equations<br />vf = vo + at(no s)<br />vf2 – vo2 = 2as(no t)<br />s = vot + ½ at2 (no vf)<br />s = vtt – ½ at2 (no vo) <br />(no a)<br />
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Components of Projectile Motion<br />The motion is in two dimensions<br />Horizontal (perpendicular to the gravitational field)<br />Vertical<br />(parallel to thegravitational field)<br />
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Components of Projectile Motion<br />Motion of the two components is independent of each other.<br />Path of the projectile is parabolic.<br />
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Labelling the diagram<br />a<br />a<br />a<br />a<br />a<br />a<br />Key here is to check what labels you’ve been asked to add – velocity? force? acceleration?<br />
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Aiming a banana above a monkeys head and he lets go of the branch<br />
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Components of Projectile Motion<br />Both the banana and the monkey accelerate at the same rate downwards.<br />Both fall the same amount below their gravity free path.<br />Banana passes over the monkey’s head.<br />Passes over by the same amount as it was originally aimed over the monkey’s head.<br />
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Components of Projectile Motion<br />What happens if you aim at the monkey?<br />
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Components of Projectile Motion<br />What happens if the banana is fired slowly?<br />
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Determining Characteristics of Projectiles<br />Step 2<br />Determining Time to Maximum Height<br />Note: Only vertical component affects the height. <br />
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Determining Characteristics of Projectiles<br />vfv = vov + at<br />a = -9.8ms-2 (i.e. acceleration is in the opposite direction to the motion when the projectile is still climbing) <br />0 = voV + at<br />vv = 0 (at maximum height)<br />
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Determining Characteristics of Projectiles<br />Step 3<br />Determining Maximum Height<br />Note: Only vertical component affects the height<br />sv = voVt + ½at2<br />a = -9.8ms-2 (i.e. acceleration is in the opposite direction to the motion)<br />Use t from Step 2<br />
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Determining Characteristics of Projectiles<br />Step 4<br />Determining range<br />Note: Only horizontal component affects range. If the ground is flat, the time in the air = 2 x time to maximum height<br />sh = voHt + ½ at2<br />a = 0 in horizontal component<br />sh = voHt<br />t = 2 x value of t in Step 2<br />
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Determining Characteristics of Projectiles<br /><ul><li>Step 5
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Horizontal component:</li></ul>sh = viHt<br />This gives distance down the range<br />
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Determining Characteristics of Projectiles<br />Vertical component: sv = vtVt + ½ at2a = -9.8 ms-2<br />This gives distance above ground. (i.e. acceleration is in the opposite direction to the motion)<br />
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Firing Projectiles Horizontally<br />What happens when you fire a projectile that has only a horizontal component?<br />
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Dropping Vertically<br />What happens if you drop a parcel from a plane?<br />
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Firing Vertically Upwards<br />What happens if you fire an object vertically upwards from a moving vehicle?<br />
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Maximum Range<br />What angle gives you the maximum range?<br />
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Different Launch Height <br />The final height may be different from the initial height.<br />How does this change the characteristics of flight? <br />The object will still follow a parabolic path.<br />It will travel further.<br />It will drop further vertically with each unit of time than if launched at the same height.<br />
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Different Launch Height<br /><ul><li>If you are throwing a ball from shoulder height which is going to land on the ground:
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A shallower angle is better.</li></li></ul><li>Effect of Air Resistance<br />Most projectiles do not follow a perfect parabolic path as there is another force besides gravity that acts on the projectile.<br />This force is due to the medium it travels through. In most cases, this is air.<br />
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Effect of Air Resistance<br />Air is a retarding force and so resists the motion.<br />Retardation depends on the size, shape and mass, speed, texture of the object.<br />It also depends on the density of the air<br />A large surface area will result in greater air resistance effects.<br />A streamlined ‘bullet’ shape will minimise the effect of air resistance.<br />
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Sport<br /><ul><li>Many sports could be used as examples – shot put, baseball, cricket, tennis and so on
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Circular Motion<br />An object moving in a circular path will have a constant speed.<br />It is continually changing direction.<br />Therefore it’s velocity is continually changing.<br />A relationship can be determined for the speed of the object.<br />
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Circular Motion Terms<br />Period<br />Is the time needed to complete one cycle/rev (in secs). The symbol T is used.<br />Frequency<br />Number of cycles/revs completed per unit time.<br />Units are Hertz (Hz)<br />f =<br />
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Circular Motion Terms<br />In uniform circular motion, the object in one revolution moves 2r in T seconds.<br />
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Centripetal Acceleration<br />A particle undergoing uniform circular motion is continually changing velocity.<br /> acceleration is changing.<br />
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Centripetal Acceleration<br />v1 = vb - va.<br />v2 = vc - vb and so on.<br />The magnitude of v1 = v2.<br />The direction is always to the centre of the circle.<br />
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Force Causing the Centripetal Acceleration<br />Any particle undergoing uniform circular motion is acted upon by an unbalanced force which is….<br />Constant in magnitude.<br />Directed towards the centre of the circle.<br />Causes the Centripetal Acceleration.<br />
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Force Causing the Centripetal Acceleration<br />Moon revolving around the Earth:<br />Gravitational Force,<br />Directed towards the centre of the Earth,<br />Holds the moon in a near circular orbit.<br />
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Force Causing the Centripetal Acceleration<br />Electrons revolve around the nucleus:<br />Electric Force,<br />Directed to centre of the nucleus,<br />Holds electrons in circular orbit.<br />
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Force Causing the Centripetal Acceleration<br />Car rounding a corner:<br />Sideways frictional force,<br />Directed towards centre of turn,<br />Force between car tyre and road.<br />If force not great enough:<br />Car skids.<br />
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Centripetal Acceleration and Friction<br />The force acts on the passenger in the car if they do not have their seat belt on.<br />Note: it is an European car.<br />
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Force Causing the Centripetal Acceleration<br />Billy can being swung.<br />Vertically or horizontally<br />The tension force between arm and can<br />causes the can to move in circular motion.<br />
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Centripetal Acceleration and the Normal Force<br />Car turns on a banked section of curved road:<br /> the chances of skidding is reduced.<br />
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Centripetal Acceleration and the Normal Force<br />
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Centripetal Acceleration and the Normal Force<br />In the vertical direction, there are 2 forces; <br /><ul><li>FNcos acting upwards and mg acting downwards. </li></ul>As there is no net vertical motion:<br /><ul><li>FNcos = mg </li></li></ul><li>Centripetal Acceleration and the Normal Force<br />For any radius curve and ideal speed, theperfect banking angle can be found.<br />
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Newton’s Law of Gravitation<br />Newton determined that a 1/d2.<br />d = distance from the centres of the objects and not the surfaces.<br />This is true for spherical objects.<br />Newton’s 2nd law also states that Fa.<br />This means that F 1/d2.<br />
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Newton’s Law of Gravitation<br />His second law also says F m.<br />As two masses are involved, Newton suggested that the force should be proportional to both masses.<br />This is also consistent with his third law. If one mass applies force on a second object, the second mass should also apply an equal but opposite force on the first.<br />
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Newton’s Law of Gravitation<br />Combining these properties, we arrive at Newton’s law of universal gravitation.<br /><ul><li> Turning this into an equality:</li></li></ul><li>Newton’s Law of Gravitation<br />Definition:<br />Between any two objects there is a gravitational attraction F that is proportional to the mass m of each object and inversely proportional to the square of the distance d between their centres.<br />
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Newton’s Law of Gravitation<br />We can find the value of g at any height above the earth’s surface.<br />
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Satellites in Circular Orbits<br />Objectswill continue to move at a constant velocity unless acted upon by an unbalanced force.<br />Newton’s first law.<br />As satellites move in a circular path, their direction (and hence velocity) is continually changing.<br />
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Satellites in Circular Orbits<br />As it is a circular orbit, <br />
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Satellites in Circular Orbits<br />This will give the orbital velocity for a satellite to remain in an orbit of r from the centre of the Earth (ie re + r) irrespective of the mass of the satellite.<br />Can you derive this equation?<br />
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Satellites in Circular Orbits<br />Speed is also given by the equation:<br />In one revolution,<br />Orbiting satellite moves a distance equivalent to the circumference of the circular path it is following.<br /> 2r <br />The time it takes for this revolution:<br /> Period (T). <br />Hence;<br />
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Artificial Earth Satellites<br />Some orbits that are preferred over others.<br />Meteorological and communication purposes. <br />Polar orbit is useful as well.<br />
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Geostationary Orbits<br />They must satisfy the following conditions:<br />They must be equatorial.<br />Only orbit in which the satellite moves in plane perpendicular to earth’s axis of rotation.<br />The orbit must be circular.<br />Must have a constant speed to match the earth’s rotation.<br />
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Geostationary Orbits<br /> The radius must match a period of 23 hrs 56 min.<br />The radius, speed and centripetal acceleration can be calculated from the period. <br /> The direction of orbit must be the same as the earth’s rotation.<br />west to east.<br />
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Low Altitude Satellites<br />200 - 3000 km above earth’s surface.<br />Used for meteorology and surveillance.<br />Smaller radius means smaller period.<br />
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Low Altitude Satellites<br />The orbit is chosen so that:<br />It passes over the same location twice each day at 12 hour intervals.<br />6am and 6pm.<br />Once in each direction.<br />As seen from the ground.<br />
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Newton’s Second Law<br />In vector form: F = ma<br />Indicates a relationship between force and acceleration.<br />The acceleration is in the same direction as the net force.<br />Implies the force on an object determines the change in velocity (aF)and<br />
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Momentum<br />Is a property of a body that is moving.<br />Vector quantity.<br />If no net force is acting on the body/bodies, momentum is defined as the product of mass and velocity.<br />
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Momentum<br />p = mv<br />Units are given as kgms-1 or sN.<br />Direction is the same as the velocity of the object.<br />
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Application of Newton II<br /><ul><li>During collisions, objects are deformed.
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For constant acceleration</li></li></ul><li>Application of Newton II<br />Ft = mvf - mvi <br />Ft = p <br />Ft= impulse of the force.<br />Impulse causes the momentum to change.<br /> An impulse is a short duration force.<br />Usually of non constant magnitude.<br />
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Application of Newton II<br />Units are the same as those for momentum<br />Kgms-1 or sN.<br />Defined as the product of the force and the time over which the force acts.<br />During collisions, t is often very small.<br />Fav is often very large.<br />
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Conservation of Momentum<br />The total momentum of all particles in an isolated system remains constant despite internal interactions between the particles.<br />
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Energy<br />The total energy in an isolated system is conserved.<br />Energy can be transferred from one object to another.<br />Energy can be converted from one form to another.<br />The units are Joules (J).<br />Is a scalar quantity.<br />Does not have a direction. <br />In collisions, total energy is always conserved.<br />
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Energy<br />The kinetic energy will not always remain constant.<br />May be converted to other forms. Could be:<br />Rotational kinetic energy<br />Sound<br />Heat.<br />
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Types of Collisions<br />Elastic collisions<br />Inelastic collisions.<br />Momentum is conserved.<br />No kinetic energy is lost.<br />Occurs on the microscopic scale.<br />Between nuclei.<br />Momentum is conserved.<br />Kinetic energy is lost.<br />All macroscopic collisions are inelastic.<br />Some collisions are almost elastic.<br />Billiard balls.<br />Air track/table gliders.<br />
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Flash Photography<br />1. Distance between successive images is a measure of speed.<br />2.Direction determined from multiple-imagephotograph.<br />3. Line joining two successive images representmagnitude and direction of velocity vector.<br />
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Flash Photography<br />To calculate distance - measure distance between successive images and adjust by the scale.<br />To calculate time - time between flashes = <br />
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Flash Photography<br />Momentum:<br />- Use velocity vector and let m1 = 1 unit and m2 is scaled accordingly.<br />- This doesn’t change the validity of the process, only the scale for the momentum vector.<br />7. Use vector diagrams for addition.<br />
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Spacecraft Propulsion<br />All vehicles move forward by pushing back on its surroundings.<br />They obey Newton’s Third Law:<br />For every action, there is a reaction.<br />
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Spacecraft Propulsion<br />Before a rocket is launched, it is stationary.<br />No momentum.<br />Total momentum after the rocket is fired:<br />must also be zero.<br />
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Spacecraft Propulsion<br />After the rocket is fired<br />Gases are ejected at high speed and, <br />As the gas has mass, <br />There is momentum acting in a direction directly opposite that in which the rocket is intended to move.<br />To conserve momentum, there must be an equal momentum acting in the direction in which the rocket moves. <br />
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Spacecraft Propulsion<br />Mass of the rocket is large compared to the gas ejected, the velocity must be…..<br />much lower.<br />As gas is ejected, mass of the rocket….<br />becomes less.<br />and the velocity….<br />becomes greater.<br />
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Spacecraft Propulsion<br />Ion Thrusters<br />Geostationary Satellites<br />Used for station keeping since 1980s<br />LEO<br />Such as Iridium mobile communications cluster<br />Deep space position control<br />Can fire ions in opposite direction to motion<br />
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Spacecraft Propulsion<br />Ion propulsion is a technique which involves<br />Ionising gas rather than using chemical propulsion<br />Gas such as Xenon<br />Heavy to provide more momentum<br />Is ionised and accelerated<br />
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Spacecraft Propulsion<br />Solar Sails<br />Converts light energy from the sun into<br />Source of propulsion for spacecraft<br />Giant mirror that reflects sunlight to<br />Transfer momentum from photons to spacecraft<br />
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Spacecraft Propulsion<br />Solar Sails have light<br />As propellant<br />Sun<br />As engine<br />Force of sunlight at the Earth<br />Is approx 4.70 N m-2 <br />
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Spacecraft Propulsion<br />Photons bounce off (or absorbed) by sail<br />During collision<br />momentum conserved<br />Small mass provides small velocity change<br />
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Spacecraft Propulsion<br />However, over time<br />Large number of photons<br />Continuous force<br />Large net force<br />eventually<br />Cannot be used to launch spacecraft<br />Still need chemical rocket<br />
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Spacecraft Propulsion<br />Reflected photons cause greater acceleration<br />than absorbed photons<br />Consider<br />for a given mass<br />p = mv<br />As velocity can change direction by 180o<br />v can double<br />p can double<br />
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Spacecraft Propulsion<br />If photon is absorbed<br />Momentum of spacecraft<br />pis =<br />pip =<br />Final momentum of system<br />
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Spacecraft Propulsion<br />If photon reflected<br />Initial momentum of spacecraft<br />ps = <br />pfp =<br />As momentum must be conserved,<br />p of system<br />psys =<br />pfs = <br />
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Spacecraft Propulsion<br />As a = v/t<br />For reflected photon<br />v can double<br />t is constant<br />Therefore a can double<br />
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