Physics 121 Tuesday January 29 2008 This is where your instructor grew up Schiphol Amsterdam Airport cemetery of ships Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Tuesday January 29 2008 Topics Course announcements Quiz Motion in two dimensions Projectile motion Problem solving strategies Circular motion Relative motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Course announcements Workshops started yesterday Monday January 28 2008 The physics laboratories started yesterday Monday January 28 2008 You are required to complete all five experiments in order to get a grade for Physics 121 If you complete less than five experiments you will get an incomplete on average 15 of the Physics 121 students get an incomplete as a results of missing laboratory experiments Homework set 1 the first one to count towards your final grade is available and is due on Saturday morning at 8 30 am Let s have a quick look at using spreadsheets Frank L H Wolfs Department of Physics and Astronomy University of Rochester Physics 121 Course announcements There will be no lecture on Thursday January 31 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 Quiz Lecture 4 The quiz today will have 4 questions 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 the two components of its motion separately For example in the case of projectile motion the gravitational acceleration only influences the motion in the vertical direction In the absence of an external force there is no acceleration in the horizontal direction and the velocity in that direction is thus constant Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions projectile motion To study projectile motion we decompose the motion into its two components Vertical motion Defines how long it will take for the projectile to hit the ground t 2v0 sin 0 g Horizontal motion During this time interval the distance traveled by the projectile is 2v0 sin 0 v0 2 x v0 cos 0 sin 2 0 R g g Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions projectile motion The equation of the range shows that the range has a maximum when sin 2 1 or 45 The range for smaller and larger angles will be smaller The difference between for example the 30 and 60 trajectories shown in the Figure is the time of flight Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions projectile motion problem solving Choose your coordinate system such that one of the axes is directed in the direction of the gravitational acceleration Where do you choose the origin of your coordinate system Determine the initial conditions e g x and y components of the velocity at time t 0 s the x and y positions at time t 0 s Calculate the time to reach the ground tgr The displacement in the horizontal direction is v0tgr Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions projectile motion problem solving The critical component of most problems is the definition of the boundary conditions for example the horizontal and vertical components of the position and the velocity The problems may differ in what you are being asked to do for example determine the range of the projectile its time of flight etc Frank L H Wolfs Department of Physics and Astronomy University of Rochester Motion in two dimensions projectile motion problem solving In general you should work with variables as long as you can Consider the trajectory problem shown in the Figure Starting point x0 0 m y0 h Starting velocity vx0 v0 cos vy0 v0 sin To calculate the range we first calculate the time t it takes to reach the ground this is just one dimensional motion in the vertical direction The range R is equal to vx0 t vx0 vy0 vy02 2hg g Check your units Now substitute your numbers to get a numerical answer Frank L H Wolfs Department of Physics and Astronomy University of Rochester Circular motion The circular motion of an object with period T can be described by the following equations x t r0 cos 2 t T y t r0 sin 2 t T The motion described by these equations is motion with constant speed v0 2 r0 T in a circle of radius r0 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Circular motion Frank L H Wolfs Department of Physics and Astronomy University of Rochester Circular motion The components of the velocity and acceleration can be obtained by differentiating x t and y t with respect to time This procedure will produce of course the same results as the graphical analysis Important facts to remember The acceleration vector points towards the center of the circle The magnitude of the acceleration is v02 r0 Frank L H Wolfs Department of Physics and Astronomy University of Rochester Relative motion The velocity of an object measured by an observer depends not only on the motion of the object but also on the motion of the observer Examples Rain appears to fall at angle when the observer is moving in the horizontal directions The relative velocity of two drivers going at 55 mph in the same direction is 0 mph Frank L H Wolfs Department of Physics and Astronomy University of Rochester Relative motion in 1D Consider two different observers A and B looking at the same car The position observations made by these observers are related in the following manner XCA XBA XCB The velocities of the car according to the two observers are related as follows VCA VBA VCB If VBA is constant then aCA aCB Frank L H Wolfs Department of Physics and Astronomy University of Rochester Relative motion in 2D and 3D The procedures to relate the observations made by different observers in 2D or 3D is similar to what we do in 1D The following relations describe the relations between the observations of observers A and B rPA rBA rPB v PA v BA v PB a PA a BA a PB Frank L H Wolfs Department of Physics and Astronomy University of Rochester Relative motion Comments An important conclusion about this discussion of relative motion is that the two observers will observe the same