Chapter 3: Vectors and Two-Dimensional Motion Saturday, January 31, 2009 11:03 AM 3.1 Vectors and their properties Vector quantity - has both direction and magnitude Scalar quantity - has a magnitude but no direction Two vectors are equal if they have the same magnitude and direction Adding the Components Method: * Resolve each vector into its components along the coordinate axes you have chosen. * Add the components of the vectors to obtain the components of the resultant vector. * Determine the magnitude and direction of the vector sum (the resultant vector) from its components using trigonometry. Trigonometric Method: * The two vectors plus their sum form a triangle which can be analyzed using trigonometry. Trigonometric analysis is useful in that it keeps you in visual contact with problem. Its limitation is that it works only if you are adding two vectors. * If the two vectors are at 90oto each other then Pythagoras' Theorem can be used to find the magnitude of the resultant vector. The angle of the resultant vector can be found using the trigonometric relations of sine, cosine, and tangent for right triangles. * When the triangle formed is not a right triangle, it still may be possible to find the magnitude of the vector sum by using the Law of cosines or the Law of sines. Law of Cosines: Law of Sines: 3.4 Motion in two dimensions Projectile Motion Horizontal component of the velocity remains constant because there is no horizontal component of acceleration The vertical component of the acceleration is equal to the free fall acceleration -g Vertical component of the velocity and the displacement in the y direction are identical to those of a freely falling body Superposition of two independent motions in the x and y direction