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Unformatted text preview: Physics 41 Lecture 4 Lecture 4: Projectile Motion Goals of this Lecture To derive the equations of motion for an object moving in two dimensions. Introduce the example of a projectile e.g. an object fired from a cannon, or a ball that is hit. To discuss the concept of relative velocity. Note that circular motion (treated in Knight sections 4.5-4.7) is deferred to Lecture 9. 4.1 Vectors in Kinematics Suppose that you are driving your car along a winding road. We assume for simplicity that the road is at, so the u1D467-direction can be ignored for now (we will add it back in later). At some particular time u1D461 1 , you are located at the end of the vector u1D45F 1 that connects your location to the origin of coordinates. We can define your position using any suitable coordi- nate system (e.g. latitude and longitude) but for this example we will use the cartesian coordinates u1D465 and u1D466 . Then u1D45F 1 = ( u1D465 1 , u1D466 1 ) Your position as a function of time defines your tra- jectory as shown opposite. At a later time u1D461 2 , you have driven to a new position, represented by vector u1D45F 2 = ( u1D465 2 , u1D466 2 ). Position vectors are useful for defining a position rel- ative to a known location. For instance you could say to a friend Ill meet you at CoHo, which is located at u1D45F = ( u1D465, u1D466 ) where the origin is the Hoover tower and where u1D465 points East and u1D466 points North. However, the magnitude and direction of a position vector defined in this way will depend on our particu- lar choice of coordinate system . For example the figure opposite shows the same tra- jectory as previously but with a different origin for the Cartesian coordinate system. Now consider the vector u1D45F which connects positions u1D45F 1 and u1D45F 2 in the first coordinate system. You can see from the figure opposite that: Neither the magnitude nor the direction of u1D45F depend on the choice of coordinate system. However the components of u1D45F may be different in different coordinate systems, if for example, one coor- dinate system is rotated relative to the other. 1 Last updated January 9, 2010 Physics 41 Lecture 4 4.1.1 The Velocity Vector In the example of you driving your car, the magnitude of u1D45F is the actual distance that you move in the time interval u1D461 = u1D461 2 u1D461 1 provided u1D461 is small (i.e. your direction of travel is approximately constant during u1D461 ). Therefore we expect u1D45F to be related to a velocity vector. Algebraically, u1D45F = u1D45F 2 u1D45F 1 = u1D45F 2 u1D45F 1 In our particular coordinate system we can write u1D45F as: u1D45F = u1D465 u1D6A4 + u1D466 u1D6A5 where u1D465 = u1D465 2 u1D465 1 and u1D466 = u1D466 2 u1D466 1 and the directions of the unit vectors are shown opposite (see Lecture 3 for further...
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