General Two and Three Dimensional Motion
Consider an object moving in three dimensions.
y

Position vector r 

r  xiˆ  yˆj  zkˆ

If the object is moving r  then
x, y, and z may be functions of time.

r
x
y
x
z
z
General Two and Three Dimensional Motion
Consider an object moving with a position given by

r  2t iˆ   4.9t 2  3 ˆj
and where the units are such that the position is in meters.


a. What are the velocity and acceleration as functions of time?

dr

v t  
dt
d

v t   2t iˆ   4.9t 2  3 ˆj
dt



d 2t  ˆ d  4.9t  3 ˆ

v t  
i
j
2
dt
dt

dv

a t  
dt
d

a t   2iˆ   9.8t  ˆj 
dt
d 2 ˆ d  9.8t  ˆ

a t  
i
j
dt
dt

v t   2iˆ   9.8t  ˆj

a t   9.8 ˆj
Notice :

vx  constant

v y  constantly changing
Notice :

uniform a y  9.8 m
Projectile Motion!
s2
General Two and Three Dimensional Motion
Consider an object moving with a position given by

r  2t iˆ   4.9t 2  3 ˆj
and where the units are such that the position is in meters.


 
b. What is y  x ? Graph the resulting motion.

x t   2t

x
t
2

yt   4.9t 2  3
 2

x
y  4.9   3
2

 x2 

y  4.9   3
 4


y  1.225x 2  3


y  1.225x 2  3