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Assignment_3_Solutions

Assignment_3_Solutions - Assignment 3 Solutions October...

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Assignment 3 Solutions October 13th, 2010 1 Problem 3.4 We start with the standard kinematics equation for an object with an acceler- ation. In this case, acceleration is gravity. h = h 0 + v 0 t + 1 2 gt 2 if we assume that the object is falling positive downward, and that h 0 and v 0 are zero and that the second ball is dropped at a time, dt , after the first then we can track the position of the two objects as they fall: 1. h 1 = 1 2 g ( t + dt ) 2 2. h 2 = 1 2 gt 2 subtracting h 2 from h 1 we get: h = h 1 h 2 = 1 2 g ( t + dt ) 2 1 2 gt 2 1 2 g ( t 2 + 2 tdt + dt 2 ) 1 2 gt 2 1 2 g ( t 2 t 2 + 2 tdt + dt 2 ) 1 2 g (2 tdt + dt 2 ) the dt 2 term stays constant while 2 tdt increases with time, therefore as time increases, so does the distance, h , between the two falling objects. answer: 1 2 g (2 tdt + dt 2 ) 2 Problem 3.8 Because we are only interested in crossing the river horizontally, we are only interested in the horizontal component of velocity. Because the river is flow- ing perpendicular to our desired direction of travel, its velocity component is unimportant in our calculation. Though the boat will drift downstream with the river, as long as we continue to paddle directly to the bank, perpendicular to the flow, none of the energy spent paddling will be wasted. answer: perpendicular to the current 1
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3 Problem 3.10 Because the object is projected o ff the roof with only horizontal velocity, we proceed initially as if the object was dropped in order to find the amount of time it took for the object to fall, we start with our standard ballistics relationship for an object with an acceleration, as before acceleration is gravity: h = h 0 + v 0 t + 1 2 gt 2 if we assum that the object starts with the roof as the zero point, and it is falling positive downward a distance, h , we get: h = 1 2 gt 2 , solving for t we get t = 2 h g so if we were to subsitute h with our two building heights H and 2 H , we get: h = H −→ T H = 2 H g and h = 2 H −→ T 2 H = 4 H g we can rewrite the 2 H case, with a little math trickery as T 2 H = 2 2 H g = 2 T H answer: T 2 H = 2 T H 2
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4 Problem 3.9 Because the object rolls o ff the table, it has no initial velocity in the y direction, V y 1. Find the time of flight of the ball
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