4 Tow-Dimensional Kinematics4-1 Motion in Two Dimensions

Recall the equation for motion in 1- dimensionvf=vi+atxf=xi+1/2(vi+vf)t Δx=xf - xi

Δx=vit+1/2at2 vav=(vi+vf)/2vf

2=vi2+2aΔx

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

),( fff yxr

yyxxr fff

),( iii yxr

yyxxr iii

),( fxfxf vvv

yvxvv fyfxf

),( iyixi vvv

yvxvv iyixi

),( yx aaa

yaxaa yx

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 t2 Δy=viyt+1/2ay t2

vfx2=vix

2+2ax Δx vfy2=viy

2+2ay Δy

same time t !

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.

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.

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 motionx-direction y-directionax=0 ay=-g=-9.8 m/s2

xf=xi+vixt vyf=viy-gt vyf

2=viy2-2gΔy

4-3 Zero Launch Angle

constant velocity along x direction

• Sketch

• Choose coordinate system

• Known & unknown quantities in x & y direction

• Apply equations

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

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

• Parabolic Pathx=vixt

yf =yi+1/2ayt2=h+1/2(-g)t2= h-1/2gt2=h-(g/2vix2)x2

t=x/vix

y=a+bx2