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Unformatted text preview: Homework2 help 2.28. Visualize: Solve: We will determine the objects velocity using graphical methods first and then using calculus. Graphically, ( ) v t v = + area under the acceleration curve from 0 to t . In this case, 0 m/s. v = The area at each time t requested is a triangle. 0 s t = ( 0 s) 0 m/s v t v = = = 2 s t = 1 ( 2 s) (2 s)(5 m/s) 5 m/s 2 v t = = = 4 s t = 1 ( 4 s) (4 s)(10 m/s) 20 m/s 2 v t = = = 6 s t = 1 ( 6 s) (6 s)(10 m/s) 30 m/s 2 v t = = = 8 s t = ( 8 s) ( 6 s) 30 m/s v t v t = = = = The last result arises because there is no additional area after 6 s. t = Let us now use calculus. The acceleration function a ( t ) consists of three pieces and can be written: 2.5 0 s 4 s ( ) 5 30 4 s 6 s 6 s 8 s t t a t t t t = - + These were determined by the slope and the y-intercept of each of the segments of the graph. The velocity function is found by integration as follows: For 0 t 4 s, 2 2 ( ) ( 0 s) ( ) 0 2.5 1.25 2 t t t v t v t a t dt t = = + = + = This gives 0 s t = ( 0 s) 0 m /s v t = = 2 s t = ( 2 s) 5 m /s v t = = 4 s t = ( 4 s) 20 m /s v t = = For 4 s 6 s, t 2 2 4 4 5 ( ) ( 4 s) ( ) 20 m/s 30 2.5 30 60 2 t t t v t v t a t dt t t t - = = + = + + = - +- This gives: t = 6 s ( 6 s) 30 m/s v t = = For 6 s 8 s, t 6 ( ) ( 6 s) ( ) 30 m/s 0 m/s 30 m/s t v t v t a t dt = = + = + = This gives: t = 8 s ( 8 s) 30 m/s v t = = 2.56. Model: The cars are represented as particles. Visualize: Solve: (a) Ann and Carol start from different locations at different times and drive at different speeds. But at time 1 t they have the same position. It is important in a problem such as this to express information in terms of positions (that is, coordinates) rather than distances. Each drives at a constant velocity, so using constant velocity kinematics gives A1 A0 A 1 A0 A 1 A0 ( ) ( ) x x v t t v t t = +- =- C1 C0 C 1 C0 C0 C 1 ( ) x x v t t x v t = +- = + The critical piece of information is that Ann and Carol have the same position at...
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This note was uploaded on 04/18/2010 for the course PHYS ? taught by Professor Zhou during the Spring '10 term at Georgia State University, Atlanta.
- Spring '10