1 HOMEWORK 1 on VECTORS: use ruler and protractor, please!!! 1. v 1 = 3m/s, E and v = 4m/s, 3 Find graphically, using scaled diagram, following vectors (both magnitude and direction): a. v = v 1 + v b. v = v 1 v c. v = v 1 + 1 v d. v = 4v 1 v. If F 1 = 3N, W and F = 6N, N find graphically and numerically the following vectors: a. F = F 1 + F b. F = F 1 + F
c. F = F 1 3 F HW VECTORS - COMPONENTS 1. Carefully construct and calculate the vertical and horizontal components of the following vectors: a. v = 3m/s, 3 v v cosθ x = v = v sinθ y v = (3m / s )cos3 x v x =.6 m/s v = (3m / s )sin 3 y v y = 1.5 m/s b. v = 3m/s, 45 c. F = 3N, 1 d. x = 3m, 1
3 e. a = 4m/s, 3. Carefully construct and calculate the magnitude and direction of the vectors that has following horizontal and vertical components: a. v x = 45 m/s, v y = 3 m/s v= v + v x y v (45m / s ) + (3m / s ) = 95m / s 3 = arctan 45 = v = 54 m/s θ θ = 34 v= 54m/s @ 34 b. v x = 3 m/s, v y = 45 m/s c. F x = - 3 N, F y = 45 N d. x x = 11 m, x y = - 15 m
4 e. a x = - 11 m/s, a y = -15 m/s 3. Find the resultant v if v 1 = 3 m/s, N; v = 4 m/s, E ; Calculate magnitude and direction of the vector v : v = v + v a. 1 b. v = v + v 1 magnitude: v = v + v direction: 1 v 1 θ = arctan v v = 3 + 4 v = 5 m/s v= 5m/s @ 37 3 θ = arctan θ = 37 4 v = v v c. 1 d. v = v v 1
5 4. Find F (magnitude F and direction ) if Draw vector F F = 6 N, ; 1 F = 6 N, 1 ; F = 6 N, 6 3 a. F = F + F b. F F + F 1 1 3 = c. F = F + F 3 5. Draw and add three vectors: F1 = (4N, ) F = (3N, 18 ) F 3 = (5N,9 ) Find resultant force! 6. A northeast and a southeast vector are being added using the parallelogram method. Which diagram below illustrates the correct method of adding two such vectors. 7. Use the following vectors to answer the questions. a. Which vectors have the same magnitude? b. Which vectors have the same direction? c. Which arrows, if any, represent the same vector?
6 HOMEWORK 3 - VECTORS - HON 1. What is the minimum number of vectors with unequal magnitudes whose vector sum can be zero? a. two b. three c. four d. five e. six. What is the minimum number of vectors with equal magnitudes whose vector sum can be zero? a. two b. three c. four d. five e. six 3. Two vectors A and B are added together to form a vector C. The relationship between the magnitudes of the vectors is given by A + B = C. Which one of the following statements concerning these vectors is true? a. A and B must be displacements. b. A and B must have equal lengths. c. A and B must point in opposite directions. d. A and B must point in the same direction. e. A and B must be at right angles to each other. 4. What is the angle between the vectors A and A when they are drawn from a common origin? a. b. 9 c. 18 d. 7 e. 36 5. Which expression is false concerning the vectors shown in the sketch? a. b. = + + = c. + + = d. C < A + B e. A + B = C 6. A jet moving at 5. km/h due east moves into a region where the wind is blowing at 1. km/h in a direction 3. north of east. What is the new velocity and direction of the aircraft relative to the ground? a. 67 km/h, 5.67 north of east c. 55. km/h, 6. north of east b. 6. km/h, 7.1 north of east d. 588 km/h, 4.87 north of east
7 7. A boat is able to move through still water at m/s. It makes a round trip to a town 3. km downstream. If the river flows at 5 m/s, the time required for this round trip is: a. 1 s b. 15 s c. s d. 3 s e. 3 s 9. An airplane makes a gradual 9 turn while flying at a constant speed of m/s. The process takes. seconds to complete. For this turn the magnitude of the average acceleration of the plane is: a = v v 1 t a.zero b. 4 m/s c. m/s d. 14 m/s e. 1 m/s 11. A body is acted upon by the two forces shown. In each case draw the one force whose effect on the body is the same as the two together. 1. A person walks 5. km due east, then 3. km due north and finally stops after walking an additional. km due north-east. How far and in what direction relative to her starting point is she? 13. A person walks 5 km east, followed by 3 km north and then another 4 km east. Where does he end up?
8 14. The displacement vector of a moving object has components (r x =, r y = ) initially. After a certain time the displacement vector has components (r x = 4, r y = 8). What vector represents the change in the displacement vector ( r = r r 1 )? 15. In physics we are often interested in finding the change in a vector. For example, consider a body whose velocity changes from to. Find vector representing the change in the velocity. 16. Figure shows the velocity vector of a particle moving in a circle with speed 1 m/s at two separate points. The velocity vector is tangential to the circle. Find the vector representing the change in the velocity vector. 17. A molecule with a velocity of 35 m/s collides with a wall as shown and bounces back with the same speed. (a) What is the change in the molecule s velocity? (b) What is the change in the speed?
9 18. A body moves in a circle of radius 3 m with a constant speed of 6. m/s. The velocity vector is at all times tangent to the circle. The body starts at A and proceeds to B and then C. Find the change in the velocity vector between A and B and between B and C. 19. Find the magnitude and direction of the vectors with components: (a) A x = 4. cm, A y = 4. cm (b) A x = 14 km, A y = -158 km (c) A x =, A y = 5. m (d) A x = 8 N, A y =.. A velocity vector of magnitude 1. m/s is horizontal. A second velocity vector of magnitude m/s must he added to the first so that the sum is vertical in direction. What is the direction of the second vector and what is the magnitude of the sum? 1.,. Find the magnitude of and and magnitude of.. Find vector (magnitude and direction) if and. 3. Find vector (magnitude and direction) if and.
1 4. Vector has a magnitude of 1 units and makes an angle of 3 with the positive x-axis. Vector has a magnitude of 8 units and makes an angle of 8 with the positive x-axis. Sketch and calculate the magnitude and directions of the vectors (a) (b) (c) 5. Vectors and have components: (A x = 3, A y = 4), (B x = 1, B y = 5). Find the magnitude and direction of the vector such that 6. Lynn is driving home from work and finds that there is road construction being done on her favorite route, so she must take a detour. Lynn travels 5 km north, 6 km east, 3 km south, 4 km west, and km south. a) Draw a vector diagram of the situation. b) What is her displacement? Solve graphically. c) What total distance has Lynn covered? 7. Avery sees a UFO out her bedroom window and calls to report it to the police. She says, The UFO moved. m east, 1. m north, and 3. m west before it disappeared. What was the displacement (how far and in what direction are they from original position) of the UFO while Avery was watching? Solve graphically and numerically. 8. Eli finds a map for a buried treasure. It tells him to begin at the old oak and walk 1 paces due west, 41 paces at an angle 45 south of west, 69 paces due north, paces due east, and 5 paces at an angle of 53 south of east. How far from the oak tree is the buried treasure? Solve graphically and numerically. 9. Ralph is mowing the back yard with a push mower that he pushes downward with a force of. N at an angle of 3. to the horizontal. What are the horizontal and vertical components of the force exerted by Ralph?
11 3. Dwight pulls his sister in her wagon with a force of 65 N at an angle of 5. to the vertical. What are the horizontal and vertical components of the force exerted by Dwight? 31. Erica and Tory are out fishing on the lake on a hot summer day when they both decide to go for a swim. Erica dives off the front of the boat with a force of 45 N, while Tory dives off the back with a force of 6. N. a) Draw a vector diagram of the situation. b) Find the resultant force on the boat. 3. Shareen finds that when she drives her motorboat upstream she can travel with a speed of only 8 m/s, while she moves with a speed of 1 m/s when she heads downstream. What is the current of the river on which Shareen is traveling? 33. Rochelle is flying to New York for her big Broadway debut. If the plane heads out of Los Angeles with a velocity of. m/s in a northeast direction, relative to the ground, and encounters a wind blowing head-on at 45 m/s, what is the resultant velocity of the plane, relative to the ground? 34. In Moncton, New Brunswick, each high tide in the Bay of Fundy produces a large surge of water known as a tidal bore. If a riverbed fills with this flowing water that travels north with a speed of 1. m/s, what is the resultant velocity of a puffin who tries to swim east across the tidal bore with a speed of 4. m/s? 35. Veronica can swim 3. m/s in still water. While trying to swim directly across a river from west to east, Veronica is pulled by a current flowing southward at. m/s. a) What is the magnitude of Veronica s resultant velocity? b) If Veronica wants to end up exactly across stream from where she began, at what angle to the shore must she swim upstream?