PROJECTILE MOTION 2-D MOTION

Projectile Motion Curve path or trajectory x and y motion at the same time Use the 4 kinematics equations x direction: x, v, t y direction: y, vf, vo, a, t 2 types Launched horizontally Launched at an angle

Launched Horizontally x-direction NO a vº = vf y-direction Δ uniformly a = g vº = 0 m/s ty = tx

Launched Horizontally

Launched Horizontally

Example 1 A stone is thrown horizontally at a speed of 5.0 m/s from the top of a cliff 78.4 m high. How long does it take the stone to reach the bottom of the cliff? How far from the base of the cliff does the stone strike the ground? What are the horizontal and vertical components of the velocity of the stone just before it hits the ground?

Example 1 Draw Diagram Make a table x y vx vº t vf a

Example 1 x y 78.4 m vx 5.0 m/s vº 0 m/s t vf a 9.8 m/s2 Solve for time y = vot + ½gt2 = ½gt2 2y 2(78.4 m) t = -------- = -------------- √ g √ 9.8 m/s/s = 4.0 s

Example 1 x y 78.4 m vx 5.0 m/s vº 0 m/s t 4.0 s vf a 9.8 m/s2 Solve for x x = vxt = (5.0 m/s) (4.0 s) = 20. m

Example 1 x y 20. m 78.4 m vx 5.0 m/s vº 0 m/s t 4.0 s vf a 9.8 m/s2 Solve for vx and vy final vx is constant = 5.0 m/s vy is changing vy = vº + gt = gt = (9.8m/s/s) (4.0 s) = 39 m/s

Example 2 A steel ball rolls with constant velocity across a tabletop 0.950 m high. It rolls off and hits the ground 0.352 m horizontally from the edge of the table. How fast was the ball rolling?

Example 2 Draw Diagram Make a table x y 0.352 m 0.950 m vx vº 0 m/s t vf a 9.8 m/s2

Example 2 x y 0.352 m 0.950 m vx vº 0 m/s t vf a 9.8 m/s2 Solve for time y = vot + ½gt2 = ½gt2 2y 2(0.950 m) t = -------- = -------------- √ g √ 9.8 m/s/s = 0.440 s

Example 2 x y 0.352 m 0.950 m vx vº 0 m/s t vf a 9.8 m/s2 0.440 s Solve for vx x vx = ------ t 0.352 m vx = ------------- 0.440 s = 0.800 m/s

Example 3 An auto, moving too fast on a horizontal stretch of mountain road, slides off the road, falling into deep snow 43.9 m below the road and 87.7 m beyond the edge of the road. How long did the auto take to fall? How fast was it going when it left the road? What was its acceleration 10 m below the edge of the road?

Example 3 Draw Diagram Make a table x y 87.7 m 43.9 m vx vº 0 m/s t vf

Example 3 x y 87.7 m 43.9 m vx vº 0 m/s t vf a 9.8 m/s2 Solve for time y = vot + ½gt2 = ½gt2 2y 2(43.9 m) t = -------- = -------------- √ g √ 9.8 m/s/s = 2.99 s

Example 3 x y 87.7 m 43.9 m vx vº 0 m/s t 2.99 s vf a 9.8 m/s2 Solve for vx x vx = ------ t 87.7 m vx = ------------- 2.99 s = 29.3 m/s

Example 3 x y 87.7 m 43.9 m vx 29.3 m/s vº 0 m/s t 2.99 s vf a Acceleration is only due to gravity.

Launched at an Angle x-direction y-direction ty = tx Solve x and y NO a vº constant y-direction Δ uniformly a = g tup = tdown ty = tx Solve x and y

Example 4 A player kicks a football from ground level with a velocity of magnitude of 27.0 m/s at an angle of 30.0º above the horizontal. Find the hang time. Find the distance the ball travels before it hits the ground. Find its maximum height.

Example 4 Draw Diagram Make a table x y vx vº t vf a 9.8 m/s2

Launched at Angle Solve for initial vx and vy vx = v cosθ = (27 m/s) cos 30º = 23 m/s vy = v sinθ = (27 m/s) sin 30º = 14 m/s x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2

Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time vy = vo + gt - 14 m/s = 14 m/s + (-9.8 m/s2) t -28 m/s = -9.8 m/s2 t t = 2.8 s

Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time y = vot + ½gt2 = ½gt2 2y 2(43.9 m) t = -------- = -------------- √ g √ 9.8 m/s/s = 2.99 s

Example 4 x y vx 23 m/s vº 14 m/s t vf a 9.8 m/s2 Solve for time y = vot + ½gt2 = ½gt2 2y 2(43.9 m) t = -------- = -------------- √ g √ 9.8 m/s/s = 2.99 s

Example 5 A rude tourist throws a peach pit horizontally with a 7.0 m/s velocity out of an elevator cage. If the elevator is not moving, how long will the pit take to reach the ground, 17.0 m below? How far from the elevator will the pit land? He throws the next pit when the elevator is at the same height but moving upward at a constant 8.5 m/s velocity. How long will it take this pit to land? How far away will this pit land?

Example 5 Draw Diagram Make a table x y 87.7 m 43.9 m vx vº 0 m/s t vf

Graphing Projectile Motion Height Horizontal Distance

Graphing Projectile Motion Height Time

Graphing Projectile Motion Vertical Speed Time

Graphing Projectile Motion Horizontal Velocity Time

Graphing Projectile Motion Horizontal Distance Time