3 Vectors and Two- Dimensional Motion

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1 May 25, Vectors and Two- Dimensional Motion Kinematics of a Particle Moving in a Plane Motion in two dimensions is easily comprehended if one thinks of the motion as being made up of two independent motions, i.e., motion in each dimension will be considered as independent of the other and then combined to make up the total motion. To represent the total motion we will introduce vectors and discuss procedures for adding and subtracting them. 3.1 Vectors and Scalars Revisted A scalar is a quantity completely specified by only a number with appropriate units. A vector is a quantity that has a number (magnitude) and a direction. Vectors are encountered whenever more than one dimension is needed to consider the quantity of interest. Examples are: displacement, velocity, and acceleration. We denote vector quantities with bold face type in this book. 17

2 Some Properties of Vectors 3.2 Some Properties of Vectors Equality of two vectors A and B: A = B means both the magnitude of A and B are the same and they are in the same direction. Their locations are not important in making them equal. They could be in entirely different locations. C A B Adding Vectors: C = A + B. Vectors can be added graphically by placing the tail (beginning point) of one vector, B, to the head (end point) of the other, A. A line connecting the tail of the first, A, to the head of the second, B, is the resultant, C, of the addition which is also a vector. Vector addition may also be accomplished by adding together components as scalars of each vector A and B, namely C x = A x + B x and C y = A y + B y. Negative of a Vector: (-B), is in the opposite direction to B but with the same magnitude. -B A -B A+(-B) Subtraction of Vectors: Subtraction is achieved by adding a vector A to a negative vector B, i.e., C = A + (- B). Displacement is a vector indicating the changes in position of a particle and is, therefore, the difference between the final and initial position vectors, i.e., r = r f - r i. Multiplication and Division of Vectors by Scalars: Multiplication of a vector by a scalar gives a new vector of different length. It changes the magnitude and hence the length of the vector, but not its direction unless the scalar is negative, in which case the vector points in the opposite direction. 18 Vectors and Two-Dimensional Motion

3 Components of a Vector 3.3 Components of a Vector A vector can be represented as the sum of its two components directed along the x and y axes. To specify these components we give their magnitudes as A x = Acosθ A y = Asinθ (3.1) where θ is the angle between a vector directed along x and A. Vector A is now resolved into its components A x and A y. The magnitude of A is denoted A or simply A (not bold). It is related to A x and A y through 2 2 A = A x + A y and A y tanθ = so θ = A x tan 1 A y A x (3.2) Adding Vectors Two or more vectors may be added algebraically by first decomposing the vectors into their components, then adding the components algebraically, so C x = A x + B x and C y = A y + B y. Of course C and its direction can be obtained from equations like Equation 3.2. (See the text for problem solving strategy when vectors are involved.) 3.4 Velocity and Acceleration in Two Dimensions The equation of motion is an expression which gives the position of a particle as a function of time. A tra- Vectors and Two-Dimensional Motion 19

4 Velocity and Acceleration in Two Dimensions jectory gives an expression for the path the particle takes in the two dimensional plane. Dealing in two or more dimensions makes the displacement, the velocity, and the acceleration variables as vectors since they may have components in each dimension. y The displacement vector is the vector difference between the final position vector, r f, and the initial position, r i, i.e., r r f - r i (3.3) r i r r f The average velocity is the displacement vector divided by the elapsed time, x v r t (3.4) and the instantaneous velocity can be computed similarly to the average velocity but the elapsed time is allowed to approach zero, which also results in the displacement vector becoming very small. v lim t 0 r t (3.5) The average acceleration is the change in velocity divided by the elapsed time t. a v t (3.6) The limit of this ratio as the elapsed time goes to zero is called the instantaneous acceleration, namely 20 Vectors and Two-Dimensional Motion

5 Projectile Motion a lim t 0 v t (3.7) θ 0 v x0 v y0 Note, whenever the velocity vector changes either in magnitude or direction, we have acceleration. 3.5 Projectile Motion A projectile, such as a baseball after it has been hit for a home-run, has vertical motion and horizontal motion. Its path describes a trajectory. It is a good example of two dimensional motion. The horizontal component of velocity is constant so x = v x0 t (3.8) whereas the vertical component of velocity is affected by gravity which accelerates the object in a downward direction. Thus, a y = - g, and v y = v y0 - gt, (3.9) y = v y0 t - gt 2 /2. (3.10) The initial velocity components can be obtained from the initial velocity and initial angle at which the particle begins its trajectory, namely v x0 = v 0 cos θ 0, and v y0 = v 0 sin θ 0. (3.11) Also we have v y 2 = v y0 2-2gy (3.12) The speed or magnitude of the velocity at any position in the trajectory is given by Vectors and Two-Dimensional Motion 21

6 Relative Velocity v = v x + v y = v x0 + v y (3.13) 3.6 Relative Velocity An object in motion observed by someone in motion may appear quite different than it does by an observer at rest or one traveling at a different velocity. A simple way of relating the various velocities is the subscript rule. The rule goes as follows. v ab means the velocity of a with respect to b. So in terms of other velocities we can write 22 Vectors and Two-Dimensional Motion v ab = v ac + v cb (3.14) Note, each velocity is a vector and the subscript starts with the letter that ended the preceding velocity subscript. Also, if the order of the subscripts is reversed there is a change in sign. So, v cd = - v dc, etc. 3.7 Concept Statements and Questions 1. How do the concepts of velocity and acceleration differ in this chapter from the ideas about these quantities developed in Chapter 2? 2. Everything explained about vectors in this chapter could be easily extended to three dimensions by simply adding another component, e.g. A z. 3. You will notice that the kinematic equations involving two dimensions are completely similar to the equations in one dimension. The total motion is obtained by simply putting the two individual motions together.

7 Hints for Solving the Problems 4. In a projectile problem the range is usually defined and figured when the vertical height, y, is equal zero. There are always two x values for every y since what goes up must come down and the x position continually changes due to the velocity in the x direction, i.e.,v x. What x corresponds to the other y = 0 value? 3.8 Hints for Solving the Problems General Hints 1. For projectile motion, remember that even though the projectile is accelerating in the vertical direction, this does not affect the speed in the horizontal direction in any way. The horizontal speed remains constant throughout the flight. Likewise, the constant horizontal speed has no effect on what is happening in the vertical direction. 2. The author s way for solving for time in Example 3.5 is perfectly OK, but there may be other ways to think about the problem and some may give an even simpler solution. For example in finding y max, it is simply, y max = gt 2 /2, where t is the time for half the trajectory, s. (Why is this so?) Of course v y0 = v 0 sinθ is the initial vertical velocity and the final velocity is the same, but in opposite direction. But these are not needed to solve for y max. The lesson is this: Look for ever simpler ways to solve the problems and try to understand the connection between the various methods and why they each work. Vectors and Two-Dimensional Motion 23

8 Hints for Solving the Problems 3. Example 3.5 is very good to study to understand the independence of motions in the x-direction and y- direction. 4.The addition and subtraction rules for relative velocity are very useful but must be applied with precision, so study the examples very carefully. Hints for Solving Selected Problems 2-6. In working these vector problems it is important to make a sketch of all of the displacements given and to identify the unknown quantity that you are solving for on the graph before you attempt the solution. This procedure prepares you for all other physics problems too Draw a sketch of the given displacements and on the sketch identify the unknown you are asked to find. Resolve the given vectors into components and combine the components by adding or subtracting as needed. Solve for the unknowns asked for. To solve for the unknown displacement in Problem 13, you can subtract the known displacements from the resultant Use the projectile equations for x and for y and eliminate the t. Thus you will have an equation involving x and y in terms of the angle of projection Remember the motion in the horizontal direction is to be treated independently from the vertical motion. The vertical speeds at the same elevation going up or down must be equal. Consider the vertical and horizontal velocities to be independent, but the 24 Vectors and Two-Dimensional Motion

9 Hints for Solving the Problems time has a common starting point so finding time from one equation and substituting into the other gives the answer Most of these problems can be solved using the relative velocity formula. If the velocities are not colinear, the equation is one of vector addition. Vectors and Two-Dimensional Motion 25

10 Hints for Solving the Problems 26 Vectors and Two-Dimensional Motion