1. Two Dimensional Kinematics
Today
Two Dimensional Kinematics

2. X and Y are INDEPENDENT! (Just like Force)
2-Dimensions
X and Y are INDEPENDENT! (Just like Force)
Break 2-D problem into two D problems 7

3. Kinematics in Two Dimensions
x = x0 + v0xt + 1/2 axt2
vx = v0x + axt
vx2 = v0x2 + 2ax x
y = y0 + v0yt + 1/2 ayt2
vy = v0y + ayt
vy2 = v0y2 + 2ay y
x and y motions are independent!
They share a common time t 21

4. Projectile Motion Demo
One marble is given an initial horizontal velocity, the other simply dropped. Which marble hits the ground first?
A) dropped
B) pushed
C) They both hit the ground at the same time
Could add y(t) equation here change to marble equation
When ball hits depends on y only!
y(t) = y0 + vyo + ½ ay t
Same for both balls! 35

5. Example: ball thrown from cliff
Use y components to find time
vo= 20 m/s
Use x velocity and time to find horizontal distance
60m

6. Shooting the Monkey... x = v0 t y = -1/2 g t2 x = x0 y = -1/2 g t2
See text: 4-3
Shooting the Monkey...
x = v0 t
y = -1/2 g t2
x = x0
y = -1/2 g t2 40

8. Shooting the Monkey... monkey! y = y0 - 1/2 g t2
Still works even if you shoot upwards!
Dart hits the
monkey!
y = v0y t - 1/2 g t2 41

9. Example: projectile Break Vo into x & y components
Find max height and distance
Vo=30m/s Δy
200 Δx
Break Vo into x & y components
Vo=30m/s
200

10. Example: projectile
Vo=30m/s Δy
200 Δx

11. How do we solve this problem?
Example: golf shot
A golfer’s ball lands 30m behind a 20m tall tree. If he hits a ball at 50m/s at an angle of 40 degrees will the ball get over the tree?
How do we solve this problem?
Find max height of ball and compare to height of tree
Find max distance of ball and compare to 30m
Find height of ball after traveling 30m in the horizontal
20m
Vo=50m/s
400
30m

12. Will the ball get over the tree?
Example: golf shot
Will the ball get over the tree?
Vo=50m/s
400
20m
Vo=50m/s
400
30m

13. Will the ball get over the tree?
Example: golf shot
Will the ball get over the tree?
20m
Vo=50m/s
400
30m

14. Summary of Concepts X and Y directions are Independent!
Position, velocity and acceleration are vectors
“Share” t
F = m a applies in both x and y direction
Projective Motion
ax = 0 in horizontal direction
ay = g in vertical direction 50