2-D Examples Questions on Homework Review – Football kick (symmetric/asymmetric) Review – Football field goal (asymmetric) Review – Range equation (symmetric) Problem 26 – Plane dropping supplies Problem 20 – Romeo and Juliet Problem 31 – Projectile from cliff Example – High fly ball (simultaneous equations).

Questions on Homework ??

Example 4 – Football Kick (symmetric) Football kicked at 20 m/s and θ = 37 V0x = 20 cos θ, V0y = 20 sin θ ax = 0 m/s2 , ay = -9.8 m/s2 Time to maximum height Maximum height Velocity at maximum height Range to maximum height Time to hit ground Distance at hit ground Velocity at hit ground

Example 5 - Football Kick (asymmetric) Football kicked at 20 m/s @ 37° - 1 m high Time to hit ground Distance to hit ground Eliminate non-physical time

Example 6 – Field Goal (asymmetric) Will it clear goalposts 3 m high, 30 m from kick? Strategy Find time to go 30 m in x direction. Find how high it is, at that time, in the y direction. If height greater than 3 m – Field Goal!

Problem 27 – Plane dropping supplies Y equation determines time in air! Voy = 0 (horizontal flight) How far will it travel in x given vo. Must drop that far in advance.

Problem 20 - Romeo and Juliet

Problem 20 - Romeo and Juliet What do you know? vy = 0, y = 4.5 m, a = - 9.8 m/s2 What tools do you have? y = ½ at2 + voyt + yo x = voxt + xo vy2 = voy2 + 2ay What can you find? Find voy =9.39 m/s2 Then find t = 0.958 s (Xword puzzle!) How can you use to find answer? Given x = 5m, ax = 0 Know t from above. vox = vx = x/t = 5.2 m/s.

Problem 31 -Projectile from Cliff

Tennis At serve, a tennis player aims to hit the ball horizontally. (a) What minimum speed is required for the ball to clear the 0.9-m-high net about 15.0 m from the server, if the ball is “launched” from a height of 2.5 m? (b) Where will the ball land if it just clears the net, and will it be “good” in the sense that it lands within 7.0 m of the net? (c) How long will it be in the air?

Water fountains Equation of Parabola 𝑦= − 1 2 𝑔 𝑡 2 +𝑣 0𝑦 𝑡 𝑥= 𝑣 0𝑥 𝑡 𝑦= − 1 2 𝑔 𝑡 2 +𝑣 0𝑦 𝑡 𝑥= 𝑣 0𝑥 𝑡 𝑡= 𝑥 𝑣 0𝑥 𝑦= − 1 2 𝑔 𝑣 𝑜𝑥 2 𝑥 2 + 𝑣 0𝑦 𝑣 0𝑥 𝑥 Equation of Parabola

Range Equation (symmetric) X and Y equations y = ½ gt2 + voyt + yo x = voxt + xo Find time when y = 0 0 = ½ gt2 + voyt t = 0, t = 2voy / g Then find x x = voxt = 2voxvoy / g x = 2vo2sinΘ cosΘ / g x = vo2sin2Θ / g Θ = 0 (min), Θ = 90 (min), Θ =45 (max) Trade-off between x and y motion Only works for symmetric!!

Example - Simultaneous Equations Sequential problems Does the football clear the goal, 3 m high and 30 m from the kick? Does the high fly clear the fence 10 m high and 95 m from home plate? Simultaneous problems When Babe Ruth hit a homer over the 12 m high right-field fence 95 m from home plate, roughly what was the minimum speed of the ball when it left the bat? Assume the ball was hit 1.0 m above the ground and its path initially made a 40° angle with the ground. 12 m = ½ (-9.8 m/s2) t2 + vo sin(40) t + 1m 95 m = vo cos(40) t t = 95 m / vo cos(40) t = 3.74 s vo == 33.16 m/s

Problem 26 – Hunter aims at target Fired horizontal – How much will miss? 𝑡= ∆𝑥 𝑣 𝑜𝑥 =.4167𝑠 𝑦= 1 2 𝑎 𝑡 2 = 1 2 9.8 𝑚 𝑠 2 0.467 𝑠 2 =0.851 𝑚 Fired at angle – What angle? X equation to range Y equation to zero Range equation (symmetric)