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
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

4. Example 5 - Football Kick (asymmetric)
Football kicked at 20 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. 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.

7. Problem 20 - Romeo and Juliet

8. Problem 20 - Romeo and Juliet
What do you know?
vy = 0, y = 4.5 m, a = 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 = 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.

9. Problem 31 -Projectile from Cliff

10. 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?

11. 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 𝑣 0𝑦 𝑣 0𝑥 𝑥
Equation of Parabola

12. 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!!

13. 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 == m/s

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