Hmwk_03_Solutions

Hmwk_03_Solutions - Physics 221 Fall 2008 Homework #3...

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1 Physics 221 Fall 2008 Homework #3 Solutions Chs. 3 and 9:1-3 Due Tues, Sept 16, 2008 3.1 A particle has a position ( x , y ) in the x - y plane versus time t given by x = (0.100 m/s 2 ) t 2 y = (0.000500 m/s 4 ) t 4 (a) At t = 10.0 s, how far is the particle from the origin of the x - y coordinate system? At that time, what is the angle of the position vector of the particle with respect to the positive x -axis? At t = 10.0 s, one has x = (0.100 m/s 2 ) t 2 = (0.100 m/s 2 )(10.0 s) 2 = 10.0 m y = (0.000500 m/s 4 ) t 4 = (0.000500 m/s 4 )(10.0 s) 4 = 5.00 m . The distance of the particle from the origin is r = x 2 + y 2 = (10.0 m) 2 + (5.0 m) 2 = 11.2 m . The angle of the particle’s position with respect to the positive x -axis is ± = arctan( y / x ) = arctan[(5.00 m) / (10.0 m)] = 26.6° . (b) Derive an expression for the path of the particle, i.e. , what is y ( x )? What is the shape of this path called? Plot the path, i.e. , plot y versus x for the interval t = 0 to t = 10.0 s. Use a spreadsheet program to make the plot if you can. From the statement of the problem, solve for t 2 from the expression for the position of x : t 2 = (10.0 s 2 /m) x . Substitute this into the expression for y ( t ) to get y = (0.000500 m/s 4 )[(10.0 s 2 /m) x ] 2 = (0.0500/m) x 2 . Thus the shape of the path is a parabola. As noted in part (a), when t = 10.0 s, one has x = 10.0 m. A plot of y ( x ) follows: 0 1 2 3 4 5 02468 1 0 y (m) x (m) Problem 3.1(b)
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2 (c) Derive expressions for the x and y components of the velocity of the particle versus time. What are the velocities of the particle at times t = 0.0 s and t = 10.0 s? Taking the time derivatives of the positions given in the problem yields: v x = dx dt = d [(0.100 m/s 2 ) t 2 ] dt = (0.200 m/s 2 ) t . v y = dy dt = d [(0.000500 m/s 4 ) t 4 ] dt = (0.00200 m/s 4 ) t 3 Thus at t = 0 we have v x = v y = 0. At t = 10.0 s we have v x = v y = 2.00 m/s. (d) Derive expressions for the x and y components of the acceleration of the particle versus time. What are the accelerations of the particle at times t = 0.0 s and t = 10.0 s?
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This note was uploaded on 01/28/2010 for the course PHYSICS 221 taught by Professor Johnson during the Fall '06 term at Iowa State.

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Hmwk_03_Solutions - Physics 221 Fall 2008 Homework #3...

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