Two Dimensional Kinematics Today Two Dimensional Kinematics

X and Y are INDEPENDENT! (Just like Force) 2-Dimensions X and Y are INDEPENDENT! (Just like Force) Break 2-D problem into two 1-D problems 7

Kinematics in Two Dimensions x = x0 + v0xt + 1/2 axt2 vx = v0x + axt vx2 = v0x2 + 2ax x y = y0 + v0yt + 1/2 ayt2 vy = v0y + ayt vy2 = v0y2 + 2ay y x and y motions are independent! They share a common time t 21

Projectile Motion Demo One marble is given an initial horizontal velocity, the other simply dropped. Which marble hits the ground first? A) dropped B) pushed C) They both hit the ground at the same time Could add y(t) equation here change to marble equation When ball hits depends on y only! y(t) = y0 + vyo + ½ ay t Same for both balls! 35

Example: ball thrown from cliff Use y components to find time vo= 20 m/s Use x velocity and time to find horizontal distance 60m

Shooting the Monkey... x = v0 t y = -1/2 g t2 x = x0 y = -1/2 g t2 See text: 4-3 Shooting the Monkey... x = v0 t y = -1/2 g t2 x = x0 y = -1/2 g t2 40

Shooting the Monkey... monkey! y = y0 - 1/2 g t2 Still works even if you shoot upwards! Dart hits the monkey! y = v0y t - 1/2 g t2 41

Example: projectile Break Vo into x & y components Find max height and distance Vo=30m/s Δy 200 Δx Break Vo into x & y components Vo=30m/s 200

Example: projectile Vo=30m/s Δy 200 Δx

How do we solve this problem? Example: golf shot A golfer’s ball lands 30m behind a 20m tall tree. If he hits a ball at 50m/s at an angle of 40 degrees will the ball get over the tree? How do we solve this problem? Find max height of ball and compare to height of tree Find max distance of ball and compare to 30m Find height of ball after traveling 30m in the horizontal 20m Vo=50m/s 400 30m

Will the ball get over the tree? Example: golf shot Will the ball get over the tree? Vo=50m/s 400 20m Vo=50m/s 400 30m

Will the ball get over the tree? Example: golf shot Will the ball get over the tree? 20m Vo=50m/s 400 30m

Summary of Concepts X and Y directions are Independent! Position, velocity and acceleration are vectors “Share” t F = m a applies in both x and y direction Projective Motion ax = 0 in horizontal direction ay = g in vertical direction 50