1. 4 Tow-Dimensional Kinematics 4-1 Motion in Two Dimensions
Recall the equation for motion in 1- dimension
vf=vi+at
xf=xi+1/2(vi+vf)t Δx=xf - xi
Δx=vit+1/2at vav=(vi+vf)/2
vf2=vi2+2aΔx

2. In 2 dimension apply the equation of motion to x & y directions

3. vfx=vix+ax t vfy=viy+ay t xf=xix+1/2(vix+vfx)t yf=xiy+1/2(viy+vfy)t
Equation of motion along x; Equation of motion along y
vfx=vix+ax t vfy=viy+ay t
xf=xix+1/2(vix+vfx)t yf=xiy+1/2(viy+vfy)t
Δx=vixt+1/2ax t Δy=viyt+1/2ay t2
vfx2=vix2+2ax Δx vfy2=viy2+2ay Δy
same time t !

4. Constant Velocity
ax=0 ay=0 vfx=vix=vx vfy=viy=vy, Δx=vxt Δy=vyt Ex1. Ball velocity is 2 m/s, in a direction 300 with horizon, the ball travel 3 m along the x, find the displacement in y.

5. Constant Acceleration
Ex. 4-2 Hummer Acceleration A hummingbird is flying in such a way that it is initially moving vertically with a speed of 4.6 m/s and accelerating horizontally at 11 m/s2 . Assuming the bird’s acceleration remains constant for the time interval of interest, find the horizontally and vertical distance through which it moves in 0.55 s.

6. 4-2 Projectile Motion: Basic Equations
Projectile motion, assumptions:
Air resistance ignored
The acceleration due to gravity is constant, downward, and has a magnitude equal to g=9.81 m/s2
The earth’s rotation is ignored.
Projectile motion
x-direction y-direction
ax= ay=-g=-9.8 m/s2
xf=xi+vixt vyf=viy-gt
vyf2=viy2-2gΔy

7. 4-3 Zero Launch Angle constant velocity along x direction Sketch
Choose coordinate system
Known & unknown quantities in x & y direction
Apply equations

8. The ball from a height 100 m with the 10 m/s horizontal velocity throws out, find how long the ball touch the ground and the horizontal displacement. y x
Vix=10m/s xi xf
h=100 m

9. Jumping a crevasse From 2. 75 m high, and jump 4
Jumping a crevasse From 2.75 m high, and jump 4.10 m the width of crevasse, find the minimal speed to land the other side. Find the land speed. vo
4.10 m
2.75 m

10. Parabolic Path
x=vixt
yf =yi+1/2ayt2=h+1/2(-g)t2= h-1/2gt2=h-(g/2vix2)x2
t=x/vix
y=a+bx2