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 = voy

2 + 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

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?

• Fired at angle – What angle?

– X equation to range

– Y equation to zero

– Range equation (symmetric)