1. 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).

2. Questions on Homework ??

3. Example 4 – Football Kick (symmetric) Football kicked at 20 m/s and θ = 37 –V 0x = 20 cos θ, V 0y = 20 sin θ –a x = 0 m/s 2, a y = -9.8 m/s 2 –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

4. 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

5. 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!

6. Range Equation (symmetric) X and Y equations – y = ½ gt 2 + v oy t + y o x = v ox t + x o Find time when y = 0 – 0 = ½ gt 2 + v oy t – t = 0, t = 2v oy / g Then find x – x = v ox t = 2v ox v oy / g – x = 2v o 2 sinΘ cosΘ / g – x = v o 2 sin2Θ / g Θ = 0 (min), Θ = 90 (min), Θ =45 (max) Trade-off between x and y motion Only works for symmetric!!

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

8. Problem 20 - Romeo and Juliet

9. What do you know? –v y = 0, y = 4.5 m, a = - 9.8 m/s 2 What tools do you have? – y = ½ at 2 + v oy t + y o –x = v ox t + x o –v y 2 = v oy 2 + 2ay What can you find? –Find v oy =9.39 m/s2 –Then find t = 0.958 s (Xword puzzle!) How can you use to find answer? –Given x = 5m, a x = 0 –Know t from above. –v ox = v x = x/t = 5.2 m/s.

10. Problem 31 -Projectile from Cliff

11. 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/s 2 ) t 2 + v o sin(40) t + 1m –95 m = v o cos(40) t –t = 95 m / v o cos(40) –t = 3.74 s –v o == 33.16 m/s

12. Problem 26 – Hunter aims at target