Physics 121 Thursday January 24 2008 Thor s Emerald Helmet Credit Copyright Robert Gendler Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Thursday January 24 2008 Topics Updated Course Information Review of motion in one dimension Motion in two dimensions Vectors Position velocity and acceleration in two and three dimensions Projectile motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Updated course information The Physics 121 workshops will start on Monday January 28 The physics 121 laboratories will start also on Monday January 28 There will be no lecture on Thursday 1 31 I will be in Europe from Wednesday 1 30 until Monday 2 4 Anyone who did not take the Diagnostic Test on Tuesday 1 22 needs to make up this test on Thursday morning 1 31 at 9 40 am in Hoyt it will take 45 minutes to complete this Diagnostic Test Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 homework set 1 Due 2 2 08 Weighted mean of v t Use spread sheets Integrate v to find d Relative velocity Error propagation Relative velocity Obtaining a from x t Relative velocity Frank L H Wolfs Complicated projectile motion Department of Physics and Astronomy University of Rochester Review of motion in one dimension Translational motion on one dimension can be described in terms of three parameters dv a constant dt The position x t units m v t v 0 a t The velocity v t units m s The acceleration a t units m s2 An important special case is the case of constant acceleration acceleration independent of time Frank L H Wolfs 1 2 x t x 0 v 0 t a t 2 Department of Physics and Astronomy University of Rochester Review of motion in one dimension dv a g dt v t v 0 g t 1 2 y t y 0 v 0 t g t 2 Note in this format g is assumed to be the magnitude of the gravitational acceleration Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two or three dimensions Vectors In order to study motion in two dimensions we need to introduce the concepts of vectors Position velocity and acceleration in two or threedimensions are determined by not only specifying their magnitude but also their direction Frank L H Wolfs Department of Physics and Astronomy University of Rochester Two dimensional motion The same displacement can be achieved in many different ways Instead of specifying a heading and distance that takes you from the origin of your coordinate system to your destination you could also indicate how many km North you need to travel and how many km East In either case you need to specify two numbers and this type of motion is called two dimensional motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Vector manipulations Any complicated type of motion can be broken down into a series of small steps each of which can be specified by a vector I will make the assumption that you have read the details about vector manipulations Vector addition Vector subtraction Frank L H Wolfs Department of Physics and Astronomy University of Rochester Vector components Although we can manipulate vectors using various graphical techniques in most cases the easiest approach is to decompose the vector into its components along the axes of the coordinate system you have chosen Frank L H Wolfs Department of Physics and Astronomy University of Rochester Vector components Using vector components vector addition becomes equivalent to adding the components of the original vectors The sum of the x and y components can be used to reconstruct the sum vector Frank L H Wolfs Department of Physics and Astronomy University of Rochester Other vector manipulations the scalar product The scalar product or dot product between two vectors is a scalar which is related to the magnitude of the vectors and the angle between them b a It is defined as aib a b cos Frank L H Wolfs Department of Physics and Astronomy University of Rochester Other vector manipulations the scalar product In terms of the components of a and b the scalar product is equal to aib ax bx i i i a y by j i j az bz k ik a b a b i i j a b a b i ik a b a b j ik x y y x x z z x y z z y ax bx a y by az bz Usually you will use the component form to calculate the scalar product and then use the vector form to determine the angle between vectors a and b Frank L H Wolfs Department of Physics and Astronomy University of Rochester Other vector manipulations the vector product The vector product between two vectors is a vector whose magnitude is related to the magnitude of the vectors and the angle between them and whose direction is perpendicular to the plane defined by the vectors The vector product is defined as c a b sin Frank L H Wolfs Department of Physics and Astronomy University of Rochester Other vector manipulations the vector product Usually the vector product is calculated by using the components of the vectors a and b i a b ax bx j ay by k az bz aybz az by i az bx ax bz j ax by aybx k Frank L H Wolfs Department of Physics and Astronomy University of Rochester Defining motion in two dimensions One Dimension Two Three Dimensions Position x Displacement x Velocity displacement per unit time Sign is equal to the sign of the displacement x Position vector r Displacement vector r Velocity vector change in the position vector per unit time The direction is equal to the direction of the displacement vector r Acceleration vector change in the velocity vector per unit time The direction is equal to the direction of the velocity difference vector v Acceleration change in velocity v per unit time Sign is equal to the sign of the velocity difference v Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions The direction of the velocity vector is tangent to the path of the object The direction of the acceleration vector is more complicated and in general is not pointing in the same direction as the velocity In non zero acceleration in two or three dimensions does not need to result in a change of the speed of an object It may only change the direction of motion and not its magnitude circular motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions When an object moves in two dimensions we can consider two components of its motion separately For example