Chapter 3. Vectors I. Vectors and Scalars 1. What type of quantity does the odometer of a car measure? a) vector; b) scalar; c) neither scalar nor vector; d) both scalar and vector. 2. What type of quantity does the speedometer of a car measure? a) vector; b) scalar; c) neither scalar not vector; d) both scalar and vector. 3. Which of the following is a vector? a) mass; b) temperature; c) speed; d) acceleration; e) time. 4. During baseball practice, a batter hits a very high fly ball, and then runs in a straight line and catches it. When comparing the displacement for the two, one sees that the? a) player's displacement is larger; b) ball's displacement is larger; c) two displacements are equal. 5. Which of the following is a scalar? a) mass; b) displacement; c) velocity; d) acceleration; e) force. II. Components of Vectors 1. Convert α = 54 to radians. 5 2. Convert β = π to degrees. 12 3. Find the î ( ĵ ) component of the vector a r if its magnitude is 15 m and the angle θ that the vector forms with i ˆ is 15 o. 1
4. An airplane is taking off with the speed of 600 km/h. Its path is linear, and it makes an angle of 17 o with the horizontal. Assuming that the speed of the airplane remains constant, find the horizontal distance that the airplane travels in 3.5 s. 5. A 12-meter ladder that stands against a wall makes an angle of 50 o with the floor. If a worker gets all the way on top of the ladder, what is the vertical distance he travels? 6. Two vectors a r and b r r are given: a = (1.2 m) iˆ+ (3.0 m) ˆj and r ˆ r r b = ( 1.0 m) i + (2. 7 m) ĵ. Find the magnitude of c = a+ b r. r r r 7. Such vectors a and b are given that a = (4.0 m) iˆ+ (6.3 m) ˆj and r ˆ r r b = (6.0 m) i + (4.9m) ĵ. Find the direction of vector c = a b r in degrees relative to i ˆ. III. Adding and Subtracting Vectors Graphically 1. Two vectors A and B are given. What would be the resultant vector C if A and B are added together? a) c) e) Hint 1/ Vectors can be added graphically without decomposing the vectors into vertical and horizontal components. In order to add the two vectors graphically, place the tail of the vector B at the head of the vector A. The resultant vector (vector C) can be found by drawing a vector from the tail of vector A to the head of vector B. 2
2. Two vectors A and B are given. Find C = A B. a) c) e) Hint 1/ Vectors can be added graphically without decomposing the vectors into vertical and horizontal components. Hint 2/ In order to subtract the two vectors graphically, treat the subtraction as addition of a negative vector A B = A + (-B). The negative vector can be represented as an arrow with the same magnitude as the original but pointing in the opposite direction. Place the tail of the vector B at the head of the vector A. The resultant vector (vector C) can be found by drawing a vector from the tail of the vector A to the head of the vector B. 3. Given three vectors A, B and C, find D = A + B + C. 3
a) c) e) Hint 1/ Vectors can be added graphically without decomposing the vectors into vertical and horizontal components. In order to add the three vectors graphically, place the tail of the vector B at the head of the vector A, then place the tail of the vector C at the head of the vector B. The resultant vector (vector D) can be found by drawing a vector from the tail of vector A to the head of vector C. 4. Given three vectors A, B and C, find D = A - B + C. a) c) e) Hint 1/ Vectors can be added graphically without decomposing the vectors into vertical and horizontal components. Hint 2/ In order to subtract two vectors graphically, treat the subtraction as addition of a negative vector A B = A + (-B). The negative vector can be represented as an arrow with the same magnitude as the original but pointing in the opposite direction. In order to find D = A - B + C, place the tail of the vector B at the head of the vector A, then place the tail of the vector C at the head of the vector B. The resultant vector (vector D) can be found by drawing a vector from the tail of vector A to the head of vector C. 4
5. Given four vectors A, B, C and D, find E = 2A + 2B + C + D. a) c) e) Hint 1/ Vectors can be added graphically without decomposing the vectors into vertical and horizontal components. Hint 2/ Vectors 2A and 2B can be represented graphically as arrows having twice the length of the respective original vectors (vectors A and B). In order to find E = 2A + 2B + C + D, place the tail of the vector 2B at the head of the vector 2A, then place the tail of the vector C at the head of the vector 2B. After that place the tail of the vector D at the head of the vector C. The resultant vector (vector E) can be found by drawing a vector from the tail of vector A to the head of vector D. IV. Position, Displacement and Velocity Vectors. 1. Find the magnitude of the displacement of a truck (car, motorcycle) that traveled 130 mi east and then 50 mi west. Assume that the x-axis is directed east. Hint 1/ Given the direction of motion and the distance that the object moved in that direction, draw two collinear vectors that describe the motion. 5
Suppose that the vector A illustrates the object s motion eastward, and the vector B illustrates the motion westward. The picture below shows vector C as the sum of vectors A and B (assuming that the magnitude of the vector A is greater than that of the vector B). 2. An airplane is flying straight north with the speed of 560 km/h. The wind that begins to blow east (from the west) has a speed of 35 km/h. Find the resultant speed (the direction of the resultant velocity vector relative to east) of the airplane. Assume that the x-axis is directed east, and the y-axis is directed north. Hint 1/ Given the magnitudes and directions of the vectors, you can draw the described vectors. Hint 2/ Suppose that vector A represents the velocity of the airplane and vector B represents the velocity of the wind. The picture below shows the resultant vector (vector C). Note that the three vectors form a right triangle. Use Pythagorean theorem to find the magnitude of the resultant velocity vector (vector C). Angle θ specifies the direction of the resultant velocity vector of the airplane relative to east. θ = 90 o α, where α = tan -1 (B/A). 3. A camp of rock climbers is located at point A. Early in the morning, the rock climbers travel 500 m to point B, the base of the cliff, then they climb straight up for 100 m and reach point C. What is the degree measure of angle θ, the angle that the displacement vector s r makes with the horizontal? 6
Use trigonometric functions to find the direction of the resultant displacement vector S: θ = tan -1 (BC/AB). 4. A motorcycle is ridden (car is driven) 12 km north, 15 km west, and then 4 km north. What is the magnitude of the displacement from the point of origin? Hint 1/ Given the magnitudes and directions of the vectors, you can draw the described vectors. Hint 2/ Add the two vectors in the north direction together and treat them as one vector. Suppose that vector A represents the two vectors directed north and that vector B represents the displacement vector directed west. Then C is the resultant displacement vector (derived by adding vectors A and B graphically). Vectors A, B, and C form a right triangle, so Pythagorean theorem can be used to find the magnitude of vector C. 5. A car is driven (motorcycle is ridden) for 34 km south, then west for 15 km, and, finally, 9 km in a direction 26 o east of north. Find the magnitude of the car s total displacement. Hint 1/ Decompose the vectors into east-west and north-south components. Hint 2/ Add or subtract the components as necessary in order to obtain two vectors: one in north-south direction and one in east-west direction. Adding these two vectors together gives a resultant vector, then the three vectors form a right triangle. Use Pythagorean theorem to find the magnitude of the resultant displacement vector. 6. The velocity vector of a particle is equal to v= ( 11.2 m / s) iˆ (15.5 m/ s) ˆj. What is the magnitude of this vector in m/s? Hint 1/ The velocity vector is already decomposed into i and j components. Hint 2/ Adding the two components graphically will give a resultant vector. The three vectors then will form a right triangle to which Pythagorean theorem can be applied to find the magnitude of the resultant vector. V. Relative Motion 7
1. The speed of a bicyclist that is riding in the direction of the wind is 10 km/h relative to the wind. If the speed of the wind is 7 km/h relative to the ground, what is the speed of the bicyclist relative to the ground? The velocity of the bicyclist relative to the ground is equal to the velocity of the bicyclist relative to the wind plus the velocity of the wind relative to the ground: v BG = v BW + v WG. 2. John (Jacob, Dave, Bryan, Chris, Terry, Kevin, Joe, Mike) runs at 10 km/h relative to the ground. He throws a rock (stone, object) in the opposite direction with the speed of 10 km/h relative to him. What is the velocity of the rock relative to the ground? (Assume that John is moving in the positive direction of the x- axis). The velocity of the rock relative to the ground is equal to the velocity of the rock relative to the boy plus the velocity of the boy relative to the ground: v RG = v RB + v BG. Note that John throws the rock in the opposite direction he is running, so the velocity of the object relative to the boy is negative. 3. The speed of each of the two trains that are approaching each other is 90 km/h relative to the ground. What is the speed of one of the trains relative to the other? The velocity of the first train relative to the second train is equal to the velocity of the first train relative to the ground plus the velocity of the second train relative to the ground: v T1T2 = v T1G + v T2G. 4. A boat is traveling downstream at 20 km/h with respect to the water. If a person on the boat walks from front to rear at 3 km/h with respect to the boat and the speed of the water is 5 km/h relative to the ground, what is the velocity of the person on the boat relative to the ground? Assume that the water is moving in the positive direction of the x-axis. Step 1: First we will find the velocity of the person relative to the water. The velocity of the person relative to the water is equal to the velocity of the person relative to the boat plus the velocity of the boat relative to the water: v PW = v PB + v BW. You will use this velocity (v PW ) in Step 2. Note that the boat is moving downstream and the person walks from the front of the boat to the rear i.e. in the opposite direction. This means that the velocity of the person relative to the boat is negative. Step 2: Now we will find the velocity of the person relative to the ground. The velocity of the person relative to the ground is equal to the velocity of the person relative to the water (from Step 1) plus the velocity of the water relative to the ground: v PG = v PW + v WG. 5. You are swimming across the river at 4 km/h relative to the water. The speed of the water is 2 km/h relative to the ground. What should be the direction of your 8
velocity relative to the water in order for your velocity vector with respect to the ground to be perpendicular to the shore? (Give the measure of the angle that is adjacent to the line perpendicular to the shore. State your answer in deg). Hint 1/ The velocity of the person relative to the ground equals to the velocity of the person relative to the water plus the velocity of the water relative to the ground: v PG = v PW + v WG. The graphical vector addition forms a right triangle. Note that α is the angle we are asked to find. sin α = v WG / v PW, thus α = sin -1 (v WG / v PW ) 6. The boat is traveling upstream on a river. The speed of the water relative to the ground is 2 m/s. What is the speed of the boat relative to the water if its velocity relative to the ground is 16 m/s an angle of 10 o upstream? Hint 1/ The velocity of the boat relative to the ground equals to the velocity of the boat relative to the water plus the velocity of the water relative to the ground: v BG = v BW + v WG (see picture), where angle α = 10 o (according to the text of the problem). Hint 2/ We need to find the velocity of the boat relative to the water (v BW ). Thus, v BW = v BG v WG. Use trigonometric functions to break v BG into horizontal (x-) and vertical (y-) components, and then subtract respective components of the vectors v BG and v WG. (Note that v WG is directed in the opposite direction of the vertical component of v BG, so assuming that the vertical component of v BG is positive, the vertical component of vector v WG is negative). The horizontal component of vector v BG equals: (v BG ) x = v BG cos α. The vertical component of v BG equals: (v BG ) y = v BG sin α. The horizontal component of vector v WG equals: (v WG ) x = 0. The vertical component of v WG equals: (v WG ) y = - v WG. Thus, in unit vector form, vector v BW is represented as follows: v BW = (v BG cos α) i + (v BG sin α) j - (- v WG ) j = (v BG cos α) i + (v BG sin α + v WG ) j. Use Pythagorean theorem to find the magnitude of the vector v BW. 9