Motion in Two or Three Dimensions - Old Dominion University

Chapter 3

Motion in Two or Three Dimensions

1

Outline

1. Position, velocity, acceleration 2. Motion in a plane (Set of equations) 3. Projectile Motion (Range, Height, Velocity, Trajectory) 4. Circular Motion (Polar coordinates, Time derivatives) 5. Relative Motion

2

1

Position, velocity, acceleration

1. Position

rr = xi^ + y^j + zk^

2. Average velocity

vrav

=

rr2 t2

- rr1 - t1

3. Instantaneous velocity

vr = lim rr = drr = dx i^ + dy ^j + dz k^ t0 t dt dt dt dt

vr = vxi^ + vy ^j + vzk^ v =

4. Average acceleration

arav

=

vr2 t2

- -

vr1 t1

5. Instantaneous acceleration ar = lim vr = dvr

t0 t dt

vx2

+

v

2 y

+

vz2

3

Part 1

Motion in a plane

4

2

Motion in Plane When Acceleration is Constant

5

Set of equations

6

3

Being practical: 2D motion with const. a

1. initial value problem:

knowing initial conditions

(x0,y0 & vx0,vy0) and acceleration

(ax,ay) one may find position (x,y) and velocity (vx,vy) at any moment in time.

2. final value problem: when some (or all) or final values are known, one may find required

vx

x v y

y

= vx0 + axt

=

x0

+

vx0t

+

1 2

axt2

= vy0 + ayt

=

y0

+

vy0t

+

1 2

ayt2

initial conditions to satisfy the final

values

3. a mixture of initial and final value

problems (just algebra :-)

7

Part 2

Projectile Motion

vx = vx0

x

= x0 + vx0t

v

y

y

= =

vy0 - gyt

y0

+

v y 0t

-

1 2

gt 2

8

4

Projectile Motion ? 2 D Example

9

Projectile Motion ? 2 D

10

5

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