Chapter 3 Vectors and Two dimensional motion Vector vs Scalar All physical quantities will be either a scalar or a vector A vector quantity has both magnitude size and direction sign A scalar is completely specified by only a magnitude size Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction 1 3 1 Vectors and their properties Example A jogger runs halfway around a circular path with a radius of 60 m What respectively are the magnitude of the displacement and the distance jogged Negative Vectors Two vectors are negative if they have the same magnitude but are in opposite directions A B A A 0 Resultant Vector The resultant vector is the sum of a given set of vectors R A B 2 Adding vectors Continue drawing the vectors tip to tail The resultant is drawn from the origin of A to the end of the last vector Measure the length of R and its angle Addition of vectors When you have many vectors just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector 3 Notes about Vector Addition Vectors obey the Commutative Law of Addition The order in which the vectors are added doesn t affect the result Vector Subtraction Special case of vector addition Add the negative of the subtracted vector A B A B Continue with standard vector addition procedure A B B A 4 3 2 Components of a Vector A component is a part It is useful to use rectangular components These are the projections of the vector along the x and y axes Components of a Vector The x component of a vector is the projection along the x axis Ax A cos q The y component of a vector is the projection along the y axis Ay A sin q Then A Ax Ay 5 Components of vectors The components are the legs of the right triangle whose hypotenuse is A A Ax Ay 2 and Position vector Displacement 2 q tan 1 3 3 and 3 4 Motion in Two dimension Ay Ax Find with respect to the positive x axis Average velocity Instantaneous velocity Instantaneous acceleration r r r1 r0 r v t r v Lim t 0 t v a Lim t 0 t 6 3 4 Motion in 2 D Using or signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion Important to remember y x X and Y are INDEPENDENT Break 2 D problem into two 1 D problems Still interested in displacement velocity and acceleration 7 Example 3 4 Two dimensional motion A ball is rolling on a horizontal surface at 5 m s It then rolls up a ramp at a 25 degree angle After 0 5 seconds the ball has slowed What is the change in velocity x direction y direction x x0 v0xt 1 2 axt2 y y0 v0yt 1 2 ayt2 vx v0x axt vy v0y ayt vix 5 m s viy 0 m s vx2 v0x2 2ax x vy2 v0y2 2ay y vfx 3 cos 25 m s vfy 3 sin 25 m s vx 3cos 25 5 2 28 m s vy 3sin 25 1 27 m s to 3 m s v vx2 v y2 2 6 m s y 3 m s x 5 m s x and y motions are independent They share a common time t 8 3 4 Projectile Motion Examples of projectile motion An object may move in both the x and y directions simultaneously It moves in two dimensions The form of two dimensional motion we will deal with is called projectile motion 9 Assumptions of Projectile Motion We may ignore air friction We may ignore the rotation of the earth With these assumptions an object in projectile motion will follow a parabolic path Rules of Projectile Motion The x and y directions of motion are completely independent of each other The x direction is uniform motion ax 0 The y direction is free fall ay g The initial velocity can be broken down into its xand y components vOx vO cos qO vOy vO sinqO 10 Motion in 2D ax 0 Motion diagram for a projectile ay g x x0 v0t y y0 v0yt 1 2 gt2 vx v0x vy v0y gt X vy2 v0y2 2g y Y v0x v0 cos 0 v0y v0 sin 0 11 Projectile at different angles Complementary values of the initial angle result in the same range The heights will be different The maximum range occurs at a projection angle of 45o Some Details About the Rules x direction ax 0 x vxot vxo vo cos qo vx constant This is the only operative equation in the x direction since there is uniform velocity in that direction 12 More Details About the Rules If the gun is pointed upward which can be seen with a widely used example shoot the monkey y direction v yo vo sin q o free fall problem a g take the positive direction as upward uniformly accelerated motion so the motion equations all hold 13 Shooting the Monkey x v0 t y 1 2 g t2 x x0 y 1 2 g t2 If the ball starts to fall at the same time as the gun shot and the gun was aimed at the ball at the beginning the bullet and the ball will drop vertically at the same rate Eventually the bullet will hit the ball Some Variations of Projectile Motion An object may be fired horizontally The initial velocity is all in the x direction vo vx and vy 0 All the general rules of projectile motion apply 14 Another example A rescue plane drops a package to stranded hikers The plane is traveling horizontally at 40 m s at a height of 100 m a Where does the package strike the ground 15