Chapter 3 Kinetics in Two or Three Dimensions, Vectors (1 week) 3.1 Vectors and scalars (p 66) 3.2 Addition of vectors, graphical methods (p 66) 3.3 Subtraction of vectors, and multiplication of a vector by a scalar (p 68) 3.4 Adding vectors by components (p 69) [ Lab. Tutorial ] 3.5 Unit vectors (p 73) 3.6 Vectors Kinematics (p 73) 3.7 Projectile motion (p 76) {Example 6 p 79} Homework : Problems 1, 5 page 91 Problems 7, 8, 9, 13 page 92 Problem 25 page 93 Problem 48 page 94

Chapter 3 Vectors 3-1 Vectors and Scalars The term velocity refers not only to how fast an object is moving but also to its direction. A quantity such as velocity, which has magnitude and direction, is a vector quantity. Other quantities that are also vectors are displacement, force, acceleration ... Each vector is represented by an arrow. The arrow is always drawn so that it points in the direction of the vector quantity it represents. The length of the arrow is drawn proportional to the magnitude of the vector quantity. Many other quantities have no direction associated with them, such as time, mass, energy and temperature. They are specified by a number and units. Such quantities are called scalar quantities.

3-2 Addition of Vectors, Graphical Methods Because vectors are quantities that have direction and magnitude, they must be added in special way: We use simple arithmetic for adding scalars. This can be used for adding vectors only when they are in the same direction. Example: a person walks 8 km east and then 6 km east, he will be 8 km+6 km=14 km east from the origin. The net or resultant displacement is 14 km to the east. If this person walks 8 km east and then walks 6 km west, he will be at 2 km east from the origin

3-2 Addition of Vectors, Graphical Methods Now the person travels 10 km east and then travels 5.0 km north. The two displacements are not in the same direction. We draw two arrows 𝑫 𝟏 and 𝑫 𝟐 to represent the two displacements The resultant displacement is represented by the arrow 𝑫 𝑹 You can measure, on the above diagram, that the person is 11.2 km from the origin at an angle 𝜃=27°: The resultant displacement vector has a magnitude of 11.2 km and makes an angle 𝜃=27° with the positive x axis. The magnitude of 𝑫 𝑹 can also be obtained using the Pythagoras theorem (only if the vectors are perpendicular) :

3-2 Addition of Vectors, Graphical Methods General rules for adding vectors graphically: This method is known as tail-to-tip method of adding vectors Which can be extended to three or more vectors, as shown below: A second way to add vectors is called parallelogram method

3-3 Subtraction of Vectors and Multiplication of a Vector by a Scalar Given a vector 𝑽 , we define the negative of this vector (− 𝑽 ) to be a vector with the same magnitude as 𝑽 but opposite in direction. The difference between two vectors 𝑉 2 − 𝑉 1 is defined as : 𝑉 2 − 𝑉 1 = 𝑉 2 + − 𝑉 1 A vector 𝑽 can be multiplied by a scalar c. We define their product so that c 𝑽 is in the same direction as 𝑽 and has magnitude 𝑐𝑉. The multiplication of a vector by a positive scalar changes the magnitude of the vector by a factor c but doesn’t modify its direction. If c is negative, the magnitude of c 𝑽 is c 𝑉 but the direction is opposite to that of 𝑽 .

3-4 Adding Vectors by Components A vector 𝑽 that lies in a plane can be expressed as the sum of two other vectors, called the components of the original vector. The components are usually chosen to be along two perpendicular directions, such as the x and y axes. The process of finding the components is known as resolving the vector into its components. The vector components are written 𝑽 𝒙 and 𝑽 𝒚 shown as arrows but dashed. The scalar components, 𝑉 𝑥 and 𝑉 𝑦 are the magnitudes of the vector components. 𝑉 𝑥 =𝑉 cos 𝜃 𝑉 𝑦 =𝑉𝑠𝑖𝑛𝜃 ⟹ 𝑉= 𝑉 𝑥 2 + 𝑉 𝑦 2 tan 𝜃 = 𝑉 𝑦 𝑉 𝑥

3-4 Adding Vectors by Components To add two vectors 𝑽 𝟏 and 𝑽 𝟐 using components, first, we resolve each vector into its components. Then, the components of the vector 𝑽 = 𝑽 𝟏 + 𝑽 𝟐 are: 𝑉 𝑥 = 𝑉 1𝑥 + 𝑉 2𝑥 𝑉 𝑦 = 𝑉 1𝑦 + 𝑉 2𝑦

3-4 Adding Vectors by Components

3-4 Adding Vectors by Components

3-5 Units Vectors Vectors can be written in terms of unit vectors. A unit vector is defined to have a magnitude exactly equal to one. In an x, y , z rectangular coordinate system, the unit vectors are called 𝐢 , 𝐣 and 𝐤 . They point respectively along the positive x, y and z axes (see figure below) A vector 𝑽 can be written in terms of its components as: 𝑽 = 𝑉 𝑥 𝐢 + 𝑉 𝑦 𝐣 + 𝑉 𝑧 𝐤 Unit vectors are helpful when adding vectors analytically (using components):

3-5 Units Vectors

3-6 Vector kinematics Suppose a particle follows a path in xy plane. At time 𝑡 1 the particule is at point 𝑃 1 , and at 𝑡 2 , it is at point 𝑃 2 . The vector 𝑟 1 is the position vector of the particule at time 𝑡 1 . And 𝑟 2 is the position vector at time 𝑡 2 In case of two or three dimensions, the displacement vector is defined as the vector representing change in position . We call it ∆ 𝑟 = 𝒓 𝟐 − 𝒓 𝟏 This represents the displacement during the time interval ∆𝑡= 𝑡 2 − 𝑡 1 We can extend our definitions of velocity and acceleration in a formal way to two- and three-dimensional motion

3-6 Vector kinematics Average velocity vector : The average velocity vector over the time interval ∆𝑡= 𝑡 2 − 𝑡 1 is defined as 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦= ∆ 𝑟 ∆𝑡 Instantaneous velocity vector : let us consider shorter and shorter time interval, we let ∆𝑡 approach zero so that the distance between points 𝑃 2 and 𝑃 1 also approaches zero. We define the instantaneous velocity vector as the limit of the average velocity as ∆𝑡 approaches zero 𝑉 = lim ∆𝑡→0 ∆ 𝑟 ∆𝑡 = 𝑑 𝑟 𝑑𝑡 The direction of 𝑉 at any moment is along the line tangent to path at that moment

3-6 Vector kinematics Average acceleration vector : The average acceleration vector over a time interval ∆𝑡= 𝑡 2 − 𝑡 1 is defined as 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑐𝑐𝑒𝑟𝑎𝑡𝑖𝑜𝑛= ∆ 𝑉 ∆𝑡 = 𝑉 2 − 𝑉 1 𝑡 2 − 𝑡 1 Instantaneous acceleration vector : is defined as the limit of average acceleration vector as the time interval ∆𝑡 is allowed to approach zero 𝑎 = lim ∆𝑡→0 ∆ 𝑉 ∆𝑡 = 𝑑 𝑉 𝑑𝑡 And is thus the derivative of 𝑉 with respect to t

3-6 Vector kinematics

3-6 Vector kinematics

3-7 Projectile motion We examine the more general translational motion of object moving through the air in two dimensions near the Earth’s surface, such as a golf ball, a thrown or batted baseball,… These are all projectile motion, which we can describe as taking place in two dimension. We will not be concerned now with the process by which the object is thrown or projected. We consider only its motion after it has been projected, and before it lands or is caught-that is, we analyze our projected object only when it is moving freely through the air under the action of gravity alone. Then the acceleration of the object is that due to gravity, which acts downward with magnitude g=9.80 m/ 𝑠 2 , and we assume it is constant.

3-7 Projectile motion Let us look at a ball rolling off the end of a horizontal table with an initial velocity in the horizantal x-direction 𝑉 𝑥 0 ( 𝑉 𝑦 0 =0, t=0, 𝑥 0 =0, 𝑦 0 =0 ) The velocity vector at each instant points is always tangent to the path Let us take y to be positive upward 𝑎 𝑦 =−𝑔 𝑉 𝑦 = 𝑉 𝑦 0 + 𝑎 𝑦 𝑡=−𝑔𝑡 y= 𝑦 0 + 𝑉 𝑦 0 t+ 1 2 𝑎 𝑦 𝑡 2 =− 1 2 𝑔 𝑡 2 In the horizontal direction, 𝑎 𝑥 =0 𝑉 𝑥 = 𝑉 𝑥 0 + 𝑎 𝑥 𝑡= 𝑉 𝑥 0 x= 𝑥 0 + 𝑉 𝑥 0 t+ 1 2 𝑎 𝑥 𝑡 2 = 𝑉 𝑥 0 t

3-7 Projectile motion

3-7 Projectile motion