Physics Worksheet Lesson 5: Two Dimensional Motion and Vectors Section: Name:

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1 Physics Worksheet Lesson 5: Two Dimensional Motion and Vectors Section: Name: 0 Introduction a. Motions take place in more than one dimension can be divided into separate motions in each dimension. This separation means that we can apply the laws that were developed for one dimension to many dimensions. b. In this lesson, we will focus at two-dimensional motion that is, motion confined to a flat surface. 1 Vector and Scalar Quantities a. Physical quantities can be categorized as either scalar or vector quantities. b. scalar quantity has magnitude only, with no direction specified. vector quantity has both magnitude and direction. Question 1 a. List five scalar quantities in physics:,,, b. List four vector quantities in physics:,, 2 Vector Representation n arrow is used to represent the magnitude and direction of a vector quantity. The length of the arrow, drawn to scale, indicates the magnitude of the vector quantity. The direction of the arrow indicates the direction of the vector quantity. Question 2 a. If the diagram represents a displacement vector of 1 km, draw a displacement vector of 3 km: 1 km b. If the diagram represents a velocity vector of 20 m/s, draw a velocity vector of 30 m/s: Tail Head e. Draw and label (a) a displacement vector with magnitude of 12.0 m and direction of 60 o, (b) a displacement vector with magnitude of 14.1 m and direction of 135 o, and (c) a displacement vector C with magnitude of 10.0 m and direction of 90 o : 20 m/s c. If 1 cm represents 10 m/s, draw a velocity vector of 20 m/s Left: d. If 1 cm represents 5 m/s 2 and the upward direction is defined as the positive direction, draw the acceleration due to gravity vector (g): Mr. Lin 1

2 3 One-dimensional vector addition: The result of adding two vectors is the sum (two vectors have same directions) or difference (two vectors have opposite directions) of the two lengths and the direction of the longer one. 5 m E. 5 m E. 10 m E. 5 m W. 10 m E. 5 m E. Question 3 a. If John walks 10 m to the right, 6 m to the left, 2 m to the right, and then 9 m to the left. What is (a) the total distance, and (b) the total displacement of John after all these movements. b. If a boat moves toward east at 25 km/hr relative to the river, while the speed of the river current is 5 km/hr to the west, what is the actual speed of the boat from an observer on the river bank? 4 Two-dimensional vector addition: The result of adding two vectors, called the resultant, is the diagonal of the parallelogram described by the two vectors. When the two vectors are perpendicular to each other, the resultant is the diagonal of the rectangle. a. 3-step (parallelogram) vector addition: i) Draw two vectors with their tail touching. ii) Draw a parallel projection of each vector with dashed lines to form a parallelogram. iii) Draw the diagonal from the point where the two tails are touching. b. Head-tail method for vector addition: 30 o R i) Choose a scale and indicate it on paper. ii) Select a starting point and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram. iii) Starting from the head of the first vector, draw the second vector to scale in the indicated direction. Label the magnitude and direction of the vector on the diagram. iv) Repeat steps (iii) for all vectors which are to be added. (The order is not important!) v) Draw the resultant from the tail of the first vector to the head of the last vector. vi) Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale. vii) Measure the direction of the resultant using the counterclockwise convention. Mr. Lin 2

3 c. Vector subtraction: One vector subtracts another vector is the same as one vector adds another negative vector. For example: R = is the same as R = + ( ). R Question 4 Find the resultants of the following vectors: a. e. b. f. c. g. F F N F d. h. If = 3 m/s, = 4 m/s, what is the magnitude of the resultant? F g Find the difference vector R = : i. j. 5 Components of Vectors a. Component: ny vector can be resolved into two component vectors at right angle to each other. These two vectors are called the components of the vector. ny vector can be resolved into vertical and horizontal components. b. Resolution: The process of determining the components of a vector is called resolution. c. Vector resolution: i) Vertical and horizontal lines are drawn from the tail of the vector. ii) rectangle is drawn that encloses the vector as its diagonal. iii) The sides of this rectangle are desired components, vector v x and v y. v v y v v y Vertical Velocity v Velocity Horizontal Velocity v x Mr. Lin 3 v x

4 d. Calculate vector components: i) The magnitude of the horizontal component v x = v cos θ. ii) The magnitude of the vertical component v y = v sin θ. Question 5 Find the horizontal and vertical components of the following vectors: a. b. V Calculate the magnitudes of the horizontal and vertical components of the following vectors: c. d. V y V = 20 m/s V y V = 10 m/s 30 o V x 40 o V x 6 Two-dimensional vector addition through vector resolution: a. Resolve all the vectors into horizontal and vertical components. b. Find horizontal component of the final resultant by adding all the horizontal components of the vectors. c. Find vertical component of the final resultant by adding all the vertical components of the vectors. d. Find the final resultant by adding the horizontal and vertical components of the final resultant. R Question 6 Find the resultant through vector resolution Mr. Lin 4

5 Question 7 a. Calculate the magnitude and direction of the resultant. y V = 20 m/s x V = 10 m/s b. Calculate the magnitude and direction of the resultant. y V = 20 m/s V = 20 m/s 30 o 30 o x c. Calculate the magnitude and direction of the resultant. y V = 20 m/s 60 o 30 o V = 20 m/s x Question 8 a. If car moves toward east at 10 m/s, and car moves toward west at 20 m/s, what s the relative velocity of car with respect to car? What s the relative velocity of car with respect to car? b. If car moves toward east at 10 m/s, and car moves toward north at 20 m/s, what s the velocity of car relative to car? What s the velocity of car relative to car? c. riverboat was to head straight north across a river with speed 3 m/s while the river current s speed is 4 m/s toward east. If the river s width is 240 m, (a) how long will it take for the boat to cross the river? (b) How far apart from the point straight across the river will the boat reach? (c) How far will the boat actually travel to cross the river? Mr. Lin 5

6 a. Relative velocity: Relative velocity is the vector difference between the velocities of two objects in the same coordinate system. For example, if the velocities of particles and are v and v respectively in the same coordinate system, then the relative velocity of with respect to (also called the velocity of relative to ) is v v. Conversely the velocity of relative to is v v. The relative velocity vector calculation for both one- and two-dimensional motion are similar. The velocity vector subtraction (v v ) can be viewed as vector addition (v + ( v )). 1 For example, in the following diagram, the velocity of the water current is 5 km/h east with respect to the observer D on the river bank. The velocities of boat, & C are 10 km/h east, 10 km/h west, and 10 km/h east respectively with respect to the water. The velocities of boat, & C with respect to the observer D are calculated and shown in the diagram. 5 km/h 10 km/h 5 km/h 15 km/h D a. For example, if the velocity of the water current is 10 km/h east with respect to the observer D and the velocity of the boat is 10 km/h north with respect to the water. The velocity of the boat with respect to the observer D is the resultant of the two vectors as shown in the diagram. C 10 km/h 5 km/h 5 km/h 15 km/h 10 km/h 5 km/h 10 km/h 10 km/h 10 km/h 45 o 10 km/h D 2 Two-Dimensional Motion Example Problems For all the following problems, assume the air resistance can be neglected. The acceleration due to gravity is g. a. riverboat was to head straight north across a river with speed v while the river current s speed is r toward east. If the river s width is d, (a) how long will it take for the boat to cross the river? (b) How far apart from the point straight across the river will the boat reach? (c) How far will the boat actually travel to cross the river? (d) If the riverboat were to go back to the original starting point with the same amount of time, what s the velocity is required in terms of speed v and angle θ? b. riverboat were to cross a river and reach the point straight north across the river in time t. If the river current s speed is r toward the east and the width of the river is d, what s the velocity of the riverboat is required in terms of speed v and angle θ? Mr. Lin 6