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Final Exam-solutions

# Final Exam-solutions - Version 048 Final Exam...

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Version 048 – Final Exam – chelikowsky – (59005) 1 This print-out should have 50 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Given: The battleship and enemy ships A and B lie along a straight line. Neglect air friction. A battleship simultaneously fires two shells (with the same muzzle velocity) at these two enemy ships. battleship A B If the shells follow the parabolic trajectories shown in the figure, which ship gets hit first? 1. need more information 2. A 3. B correct 4. both at the same time Explanation: The time interval for the entire projectile motion is given by t trip = t rise + t fall = 2 t rise , where t rise is the rising time from 0 to the maximum height, and t fall the falling time from h to 0. In the absence of air resistance t rise = t fall , h = 1 2 g t 2 fall , or t trip = 2 radicalBigg 2 h g . So the smaller is h , the smaller is t trip . In other words, enemy ship B will get hit first. 002 10.0 points When a metal ball of unknown mass M is suspended from a spring of unknown force constant k , the spring’s equilibrium length increases by Δ L e . And when the ball is out of equilibrium, it oscillates up and down with a period T . Find Δ L e . The acceleration of gravity is 9 . 8 m / s 2 and the period is 0 . 412 s. 1. 12.303 2. 6.76406 3. 9.54223 4. 1.704 5. 12.1289 6. 4.21367 7. 8.06522 8. 4.52608 9. 15.258 10. 13.7039 Correct answer: 4 . 21367 cm. Explanation: Let : g = 9 . 8 m / s 2 , and T = 0 . 412 s . Let L be the length of the free spring with- out the ball. When a ball is suspended in equi- librium from the spring, the spring lengthens by Δ L e to generate the tension force F S e = k Δ L e = Mg, so (1) Δ L e = Mg k . (2) Now consider the oscillating ball. When the ball is at height y above the equilibrium point, the spring is stretched by Δ L ( y ) = Δ L e y (3) and has tension F S ( y ) = k Δ L ( y ) = k L e y ) = Mg k y. (4) Consequently, the net vertical force on the ball is F net y = F S Mg = k y, (5) which makes the ball oscillate with angular frequency ω = radicalbigg k M (6)

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Version 048 – Final Exam – chelikowsky – (59005) 2 and period T = 2 π ω = 2 π radicalbigg M k . (7) We do not know the ball’s mass M nor the spring’s force constant k , but given the oscillation period T we may find the ratio M k = parenleftbigg T 2 π parenrightbigg 2 , so (8) Δ L e = Mg k = gT 2 4 π 2 = (9 . 8 m / s 2 )(0 . 412 s) 2 4 π 2 = 4 . 21367 cm . (9) 003 10.0 points Given: Two vectors vector A = A x ˆ ı + A y ˆ and vector B = B x ˆ ı + B y ˆ  , where A x = 2, A y = 9, B x = 2, and B y = 10. Find the z component of vector A × vector B . 1. 86 2. 2 3. 38 correct 4. 94 Explanation: It follows from the definition of cross prod- uct that ( vector A × vector B ) z = A x B y A y B x = ( 2) (10) (9) (2) = 38 . 004 10.0 points The distance between the sun and the Earth is about 1 . 4991 × 10 11 m. Express this distance in kilometers.
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