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Unformatted text preview: Problem Set 3: Problem 2.
Problem: The sum of the aerodynamic drag force and the road-friction force, D, on an automobile coasting along a horizontal surface is D = m g + v 2 /L , where m is the automobile's mass, v is velocity, is dimensionless friction coefficient, g is gravitational acceleration and L is a characteristic length. (a) Assuming the automobile's initial velocity is vo when it starts coasting, find the distance, d, that the car coasts until it comes to rest. Express your answer as a function of g, L, and vo . (b) Find the time, tf , as a function of g, L, and vo that it takes for the automobile to come to rest. HINT: Make use of the fact that 1 dv = tan-1 c + v2 c v c Solution: Newton's Second Law tells us that ma = -D a = -g - v2 L This is the starting point for both Parts (a) and (b). (a) Using the fact that a = vdv/dx, we have v Thus, we find
0 v2 dv = -g - dx L Lvdv = -dx gL + v 2 L
vo vdv =- gL + v 2 d dx
0 gL 2 gL + vo L n gL + v 2 2 v=0 v=vo = -d Evaluating the left hand side yields L n 2 Therefore, the total distance traveled is d= L v2 n 1+ o 2 gL = -d (b) Turning to the time elapsed, we use the fact that a = dv/dt. Hence, v2 dv = -g - dt L Ldv = -dt gL + v 2 ...
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- Spring '06