Projectile Motion

Projectile Velocities

Sammy ACT: Sammy Sosa clobbers a fastball toward center- field. The ball is hit 1 m (y o ) above the plate, and its initial velocity is 36.5 m/s (v o ) at an angle of 30 o (  ) above horizontal. The center- field wall is 113 m (x) from the plate and is 3 m (h) high. çWhat time does the ball reach the fence? çDoes Sammy get a home run? çIn other words, will the ball clear the fence?  vovovovo h x y0y0

Sammy ACT:  vovovovo y x y0y0 l Choose y axis up. l Choose x axis along the ground in the direction of the hit. l Choose the origin (0,0) to be at the plate. l Say that the ball is hit at t = 0, x = x 0 = 0 y x

Sammy ACT Equations of motion are: Horizontal Vertical v x = constant v y = v y0 - gt x = v x t y = y 0 + v y0 t - 1 / 2 gt 2 Remember that the horizontal and vertical motions are Independent of one another, but they share the time. Variables: v 0 = 36.5 m/s t = ???  = 30 o is y > h = 3 m??? y 0 = 1 m x = 113 m

Sammy... l Use geometry and trigonometry to figure out v x and v y0 : y x g  v0v0v0v0 vxvx v y0 Find v x = v cos . and v y = v sin . y0y0 v x = 36.5 cos(30 o ) m/s = 31.6 m/s v y0 = 36.5 sin(30 o ) m/s = m/s 36.5 m/s 30 0

Sammy ACT l We can use the horizontal motion equation to determine the time to reach the wall: l x = v x t l t = x / v x l t = (113 m) / (31.6 m/s) = 3.58 s l We have an equation that tells us y(t) = y 0 + v y0 t + a t 2 / 2 y(t) = (1.0 m) + (18.25 m/s)(3.58 s) - (0.5)(9.8 m/s 2 )(3.58 s) 2 m = ( ) m = 3.5 m ç Since the wall is 3 m high, Sammy gets the homer!!

Projectile Range Equations of motion are: Horizontal Vertical v x = constant v y = v y0 - gt x = v x t y = y 0 + v y0 t - 1 / 2 gt 2 l Equations for x and y can be combined (click here to see the derivation) to yield the following equation:click here to see the derivation

Range of Projectile

Golf Range ACT l A golf ball is chipped at an angle of 46 o and with a speed of 5.4 m/s. How far does it travel to the nearest tenth of a meter?

Projectile Launched at Angle

Feeding the Monkey (Banana Gun) l Where does the zookeeper aim if he wants to hit the monkey? ( He knows the monkey will let go as soon as he shoots ! )

Feeding the Monkey... l If there were no gravity, simply aim at the monkey r = r 0 r =v 0 t

Feeding the Monkey... rvg r = v 0 t - 1 / 2 g t 2 l With gravity, still aim at the monkey! rg r = r / 2 g t 2 Banana hits monkey!