PHYS Summer Professor Caillault Homework Solutions

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PHYS 1111 - Summe 2007 - Pofesso Caillault Homewok Solutions Chapte 3 13. Pictue the Poblem: The whale dives along a staight line tilted 20.0 below hoizontal fo 150 m as shown in the figue.

1 PHYS Summe Pofesso Caillault Homewok Solutions Chapte Pictue the Poblem: The whale dives along a staight line tilted 20.0 below hoizontal fo 150 m as shown in the figue. Stategy: Resolve the whale s displacement vecto into hoizontal and vetical components in ode to find its depth y and its hoizontal tavel distance x. Solution: 1. (a) The depth is given by y : 2. (b) The hoizontal tavel distance is given by x : y = sinθ = ( 150 m)sin ( 20.0 ) = 51 m x = cosθ = ( 150 m)cos( 20.0 ) = 140 m = 0.14 km Insight: Note that both answes ae limited to two significant figues, because although 20.0 has thee, 150 m has only two significant figues.

2 19. Pictue the Poblem: The vectos involved in the poblem ae depicted at ight. Stategy: Use the vecto component method of addition and subtaction to detemine the components of each combination of A and B. Once the components ae known, the length and diection of each combination can be detemined faily easily. Solution: 1. (a) Detemine the components of A + B : A + B = 5 ( ) ŷ + 10 ( ) ˆx = 10ˆx 5ŷ 2. Find the magnitude of A + B : A + B = ( 10 ) 2 + ( 5) 2 = 11 units 3. Detemine the diection of A + B, measued counteclockwise fom the positive x axis. 4. (b) Detemine the components of A B : θa+ B = 5 10 = 27 o 333 A B = 5 ( ) ŷ 10 ( ) ˆx = 10ˆx 5ŷ 5. Find the magnitude of A B : A B = ( 10 ) 2 + ( 5) 2 = 11 units 6. Detemine the diection of A B, measued counteclockwise fom the positive x axis. 7. (c) Detemine the components of B A : θa B = 5 10 = = 207 B A = 10 ( ) ˆx 5 ( ) ŷ = 10ˆx + 5ŷ 8. Find the magnitude of B A : B A = ( 10 ) 2 + ( 5) 2 = 11 units 9. Detemine the diection of B A, measued counteclockwise fom the positive x axis. θb A = 5 10 = 27 Insight: This poblem is simplified by the fact that A and B have only one component each, but a simila appoach will wok even with moe complicated vectos. Notice that you must have a pictue of the vectos in you head (o on pape) in ode to coectly intepet the diections in steps 3, 6, and Pictue the Poblem: The vectos involved in the poblem ae depicted at ight. Stategy: Use the infomation given in the figue to detemine the components of vectos A, B, and C. Then add the components. Solution: 1. Add A the x component of x = ( 1.5 m)cos( 40 ) = 1.1 m each vecto: B x = ( 2.0 m)cos( 19 ) = 1.9 m C x = ( 1.0 m)cos ( ) = 0.91 m ( A + B + C ) = 2.1 m x

3 2. Add the y component of each vecto: A y = ( 1.5 m)sin 40 B y = ( 2.0 m)sin 19 C y = ( 1.0 m)sin A + B + C ( ) = 0.96 m ( ) = 0.65 m ( ) = 0.42 m ( ) = 0.74 m y 3. Expess the sum in unit vecto notation: A + B + C = ( 2.1 m) ˆx + ( 0.74 m) ŷ Insight: In this poblem the vecto component method of addition is much quicke than the gaphical method. 31. Pictue the Poblem: The displacement vectos ae depicted at ight. Noth is in the y diection and east is in the x diection. Stategy: Sum the components of the vectos in ode to detemine A + B. Multiply that vecto by 1 in ode to evese its diection. Then find the magnitude and diection of the evesed vecto. Solution: 1. (a) Add the two displacement vectos: 2. Multiply by 1 in ode to evese the diection of the net displacement and bing the cat back home: A + B = 72 m ( ) ˆx + ( 120 m) ŷ ( A + B ) = ( 72 m) ˆx + ( 120 m) ŷ 3. Find the magnitude of the desied displacement: ( A + B ) = ( 72 m) 2 + ( 120 m) 2 = 140 m 4. Find the diection of the desied displacement: θ = tan m 72 m = 59 = 59 south of east 5. (b) Vecto addition is independent of the ode in which the addition is accomplished. The initial displacement is the same, so thee is no change in the displacement fo the homewad pat of the tip. Insight: In this poblem we could claim the cat s initial displacement is a single vecto with the given components. The answes wouldn t change, but it would simplify the solution a little bit. 43. Pictue the Poblem: The vectos involved in this poblem ae depicted at ight.

4 Stategy: Let v yw = you velocity with espect to the walkway, v wg = walkway s velocity with espect to the gound, and add the vectos accoding to equation 3-8 to find v yg = you velocity with espect to the gound. Then find the time it takes you to tavel the 85- m distance. Solution: 1. Find you velocity with espect to the walkway: 2. Apply equation 3-8 to find you velocity with espect to the gound: 3. Now find the time of tavel: v yw = Δx Δt ˆx = 85 m 68 s ˆx = ( 1.25 m/s) ˆx v yg = v yw + v wg = ( 1.25 m/s) ˆx + ( 2.2 m/s) ˆx = ( 3.45 m/s) ˆx t = Δx = 85 m v yg 3.45 m/s = 25 s Insight: The moving walkway slashed you time of tavel fom 68 s to 25 s, a facto of 2.7! Note that we bent the significant figues ules a little bit by not ounding v yw to 1.3 m/s. This helped us avoid ounding eo.

5 55. Pictue the Poblem: The thee-dimensional vecto is depicted in the diagam at ight. Stategy: Detemine the z component of A by applying the cosine function to the ight tiangle fomed in the z diection. Then find the pojection of A onto the xy plane (A sin 55 ) in ode to find the x and y components of A. Solution: 1. Find the z component of A : A z = ( 65 m)cos55 = 37 m 2. Find the pojection onto the xy plane: A xy = Asin55 = ( 65 m)sin55 3. Find the x component of A : A x = ( 65 m)sin55 cos 35 = 44 m 4. Find the y component of A : A y = ( 65 m)sin55 sin 35 = 31 m Insight: A knowledge of ight tiangles can help you find the components of even a thee dimensional vecto. Once the components ae known, then addition and subtaction of vectos become staightfowad pocedues.

PHYS Summer Professor Caillault Homework Solutions. Chapter 9

PHYS Summer Professor Caillault Homework Solutions. Chapter 9 PHYS - Summe 007 - Pofesso Caillault Homewok Solutions Chapte 9 3. Pictue the Poblem The owne walks slowly towad the notheast while the cat uns eastwad and the dog uns nothwad. Stategy Sum the momenta