Physics 215 – Fall 2014Lecture Welcome back to Physics 215 Today’s agenda: Velocity and acceleration in two-dimensional motion Motion under gravity -- projectile motion Acceleration on curved path

Physics 215 – Fall 2014Lecture Current homework assignment HW2: –Ch.2 (Knight textbook): 48, 54 –Ch.3 (Knight textbook): 28, 34, 40 –Ch.4 (Knight textbook): 42 –due Wednesday, Sept 10 th in recitation

Physics 215 – Fall 2014Lecture Exam 1: next Thursday (9/18/14) In room 208 (here!) at the usual lecture time Material covered: –Textbook chapters –Lectures up through 9/16 (slides online) –Wed/Fri Workshop activities –Homework assignments Work through practice exam problems (posted on Blackboard) Work on more practice exam problems next Wednesday in recitation workshop

Physics 215 – Fall 2014Lecture Motion under gravity a y = -g v y = v 0y - gt y = y 0 + v 0y t - (1/2)gt 2 a x = 0 v x = v 0x x = x 0 + v 0x t x y v0v0  v 0y = v 0 sin(  ) v 0x = v 0 cos(  ) Projectile motion...

Physics 215 – Fall 2014Lecture Projectile question A ball is thrown at 45 o to vertical with a speed of 7 m/s. Assuming g=10 m/s 2, how far away does the ball land?

Physics 215 – Fall 2014Lecture A battleship simultaneously fires two shells at enemy ships. If the shells follow the parabolic trajectories shown, which ship will be hit first? A. A B. Both at the same time C. B D. need more information

Physics 215 – Fall 2014Lecture Projectile motion R : when is y=0 ? t[v y1 -(1/2)gt] = 0 i.e., T = (2v)sin  /g  R (x-eqn.) max (y-eqn.)  h max (y-eqn.) y x

Physics 215 – Fall 2014Lecture Maximum height and range

Physics 215 – Fall 2014Lecture Motion on a curved path at constant speed Is the acceleration of the object equal to zero?

Physics 215 – Fall 2014Lecture Velocity is tangent to path O sIsI sFsF ss v =  s/  t lies along dotted line. As  t  0 direction of v is tangent to path

Physics 215 – Fall 2014Lecture Motion on a curved path at constant speed

Physics 215 – Fall 2014Lecture Subtracting vectors vFvF -vI-vI vv same as vFvF vIvI Recall that v F + (-v I ) =  v vv

Physics 215 – Fall 2014Lecture For an object moving at constant speed along a curved path, the acceleration is not zero.

Physics 215 – Fall 2014Lecture For which of the following motions of a car does the change in velocity vector have the greatest magnitude? (All motions occur at the same constant speed.) A.A 90° right turn at constant speed B.A U-turn at constant speed C.A 270° turn on a highway on-ramp D.The change in velocity is zero for all three motions.

Physics 215 – Fall 2014Lecture A. B. C.D. A car moves along the path shown. Velocity vectors at two different points are sketched. Which of the arrows below most closely represents the direction of the average acceleration?

Physics 215 – Fall 2014Lecture A. B. C.D. A child is riding a bicycle on a level street. The velocity and acceleration vectors of the child at a given time are shown. Which of the following velocity vectors may represent the velocity at a later time? a

Physics 215 – Fall 2014Lecture A biker is riding at constant speed clockwise on the oval track shown below. Which vector correctly describes the acceleration at the point indicated?

Physics 215 – Fall 2014Lecture Biker moving around oval at constant speed As point D is moved closer to C, angle approaches 90°.

Physics 215 – Fall 2014Lecture Summary For motion at constant speed, instantaneous acceleration vector is perpendicular to velocity vector Points ``inward’’ What is the magnitude of the acceleration vector?

Physics 215 – Fall 2014Lecture Acceleration vectors for ball swung in a horizontal circle at constant speed v What is the magnitude of the acceleration?  a  = v 2 /R   R v1v1 v2v2 v1v1 v2v2

Physics 215 – Fall 2014Lecture Acceleration of object moving at constant speed on a circular path: Acceleration depends on radius of circle.

Physics 215 – Fall 2014Lecture Two cars are moving at different constant speeds on a curved road. One after the other, they are passing the same point on the road: Car A at 18 mph; car B at 36 mph. If car A’s acceleration is 2 m/s 2, car B’s acceleration is: A.1 m/s 2 B.2 m/s 2 C.4 m/s 2 D.8 m/s 2

Physics 215 – Fall 2014Lecture What is the magnitude of the acceleration of an object moving at constant speed if the path is curved but not a circle? “r” is the radius of curvature of the path at a given point

Physics 215 – Fall 2014Lecture Acceleration vector for object speeding up from rest at point A ?

Physics 215 – Fall 2014Lecture What if the speed is changing? Consider acceleration for object on curved path starting from rest Initially, v 2 /r = 0, so no radial acceleration But a is not zero! It must be parallel to velocity

Physics 215 – Fall 2014Lecture Acceleration vectors for object speeding up: Tangential and radial components (or parallel and perpendicular)

Physics 215 – Fall 2014Lecture Reading assignment Circular motion 4.5 – 4.7 in textbook