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ame301 spring 11

# ame301 spring 11 -

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Unformatted text preview: NAME: Exam One: Open Book, Open Notes, 80 Minutes February 14, 2011 1. (30 Points) KINEMATICS SCORE: AME 301: Spring 2011 D. C. Wilcox A tank is approaching a Humber as shown. The Humber is moving at constant speed V to the right and the tank is moving with constant speed 1 V to the left. When the distance between the 2 √ two vehicles is L, the tank fires a large artillery shell at the Humber with speed 3 V relative to its cannon. At the same instant, the Humber fires a precision laser-targeted shell buster with speed v relative to its cannon. The goal is to destroy the artillery shell before it hits the Humber. Both vehicles’ cannons are at an angle φ = 30o to the horizontal. You can assume the shells from both cannons are at z = 0 when they are fired, and you can neglect effects of friction on the projectiles. (a) In order for the shell buster to destroy the tank’s artillery shell before it lands, what must the ratio v/V be? (b) At what time will the two projectiles collide? Express your answer in terms of V and L. . . z .................................................................................................................................................... . . . .. .. .......................................................................... ......................................................................... .. .. . . . . . . . . . . . ............. . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . .. . . . . . . . . .. . . . . . .. . .. . . .. . . . . . . . ..... . .. . . .. . . . . ...... . .. . . . . .. ..... . . . .. .. ..... .. . . . .. . ..... . . .. . .. .... ... . .. . ...... . .... . . . . ...... . . . ..... . .... .. . .... . .... .... . . .... . ..... . . . .. .. . . .. . .... .. ... . ....... .. . . . ....... . . . ....... . . . .. . .................................... . ..................................... . .. . .............................. .................. ........... . . . L v √ 3V φ . . . . . . . . . . . . . . . . . . . . . . ... .. . .. . . g φ x 1 .. . .............. .............. . . . V ................... V 2 ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ..................................................................... ..... ..... ..... ..... ..... ..... ...... .......... .... .... .... ...................................................................................................................................................................................................................................................................................................................................................................... .... .... .... .... .... .... .... .... .... ..... .... ................... ............. ................................. ...... ............................................. . . . .. .. .. .. .. .. .. .. .. .. .. ....... ...... ....... ..... .......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .......... ... ... ... ... ... ... ... ... .... ... .... . .......... . .......... . .... ...................................................................................... ........................................................... ............................................. ...... .................................................... ...... ............................................................................... 2. (30 Points) RECTILINEAR MOTION In general, the wave drag on a sailboat of mass m is given by D = 1 ξ (vh + 2v ) v 2 , where ξ is a 2 constant of dimensions mass·time/length2 and vh is a constant velocity scale. The thrust provided by the boat’s sails is T = 3 ξ vh v2 . The sailboat travels in a straight line along the x axis and v = 1 vh 2 5 at x = 0. Determine the sailboat’s speed as a function of ξ , vh , x and m. HINT: Using partial fractions... 1 1 = v (vh − v ) vh 1 1 + v vh − v D ............... . .............. . . ......................... . ... .. ......... . . . ................... .................. .... ... ................ ................ ..................... .. .................... .. ............ ............ ... . ... .. T 1 3. (40 Points) CURVILINEAR MOTION An MIT student is the proud owner of a classic 1954 Mercury. On a morning when the temperature in Cambridge, Massachusetts is −2o F, the voltage output of the car’s 6-volt battery has dropped to just under 4 volts, which is insufficient to start the car. In order to be on time for his dynamics class, he has decided to push start the Mercury. He moves the vehicle out to the street in order to take advantage of the hill he lives on. The hill is approximately circular near the top with radius R. (a) Derive a differential equation for the Mercury’s acceleration, dv/dt, near the top of the hill as a function of R, θ, the car’s speed, v , gravitational acceleration, g , and the coefficient of rolling friction between the car’s tires and the street, µr . Your equation should not involve the normal force of the street on the car. (b) Verify that, in the absence of rolling friction, your equation simplifies to dv/dt = g cos θ. Noting that dv/dt = (dθ/dt) · (dv/dθ), solve for v as a function of g , R and θ. Assume the car is at rest at the top of the hill. z ..... . . . . . . . . . . . . . . ........ . . ........ Hill . . . ...... .... . . .............. . . .. ...... . . .... . .. .. . . . .... .. . .... . . . . . . ... ... ... ... . . . ... .. ... .. . . ... . . .. . . . .. .... ... . . . ..... ..... . . . . . . ..... ... .... . . ..... .... .... .. . . ... . ... ... . .... . . . . ... ... . . ... ... . . . ... . ... . . . ... ... . ... . ... . .. . ... ... ... . .. . ... ... . ... . ..... ’54 Mercury ... ... . . . . .. .... .... . .... .. . .. .. .. . ...... . . ..... . ....................................................................... . ........................................................................ ... ... . . . R θ v c s g = −g k x 2 ...
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