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Introduction to Motion and Acceleration

85 the units of speed have dimension lt so if we were

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Unformatted text preview: speed have dimension [L]/[T] so if we were measuring distance in feet and time in seconds, the speed would be approximately 5.85 feet/second. Projectile Motion Of all motions in space, projectile motion is the one that everyone is familiar with because it represents the motion that an object experiences when it moves under the influence of the Earth’s gravitational force and possibly other resistive forces such as air drag. The simplest motion is the following: above the ground with an initial An projectile of mass is fired from some initial height . The forces acting on the object are the Earth’s gravitational force , , velocity 0 and a drag force which always acts in the direction opposite to motion and is such that where k is a non-negative constant. We treat the general case in an optional section at the end of these notes and limit ourselves to the more restrictive case of drag-free motion. If there is no drag force ( parallel to and take 0 . Let us set up a coordinate system in which the force of gravity is 32 ft/sec2. 0 , , 32 We wish to describe the projectile’s motion. If there is no drag force, then the only force is that of gravity. The acceleration vector is therefore 32 We can first reproduce the velocity vector: , , 32 , , 32 6 Combining the vectors, , , 32 In unit-vector notation, 32 Notice that the x and y components of the velocity vector remain constant. That is because there are no forces in these directions. The z-component, on the other hand, changes according to 32 . Now we reproduce the position vector: , , 0,0, 32 0,0, 16 Combining the vectors, 16 Thus, the parametric equation of the trajectory are 16 Often problems are posed in two dimensions in the usual x-y plane with the force of gravity acting vertically down along the y-axis. The equations that will ensue in that set up are: 16 In the more general cases where drag is not present, it is useful to write the position vector in general terms: 1 0 0 2 The constant is the acceleration an object experiences due to the force of gravity. Its value is approximately 9.8 m/s2 in the Metric System and, as was stated earlier, 32 ft/s2 in the English system. This one formula can be used to solve any drag-free situation. Example 5 Suppose a projectile is fired from ground level with a speed of 400 ft/sec at an angle of 45° with the horizontal. Determine the position ve...
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