WAN HESS, ☒HNXIEEMYAATT 118$ HANK☒ DDYIINVAAMY118$
motion in two or three dimension
define coordinate system ( define origin & the ✗ -
, y -
,
and z-axis directions )
In 3D, determine position ( r ) at any moment in time
any position vector can be written as the sum of three vectors in the direction of each axis
trajectory : the path an object travels over time Fct) =
✗ (t ) it yet )j + 2 A) it
this requires you to find functions of change in position relative to each single axis over
time ( ✗ (t ) , y (t ) , 2- (t ) )
motion along a straight line can be used for each axis separately (axes are independant )
displacement and average velocity
displacement F' Ñ (ta) Flt ) : = -
.
= ri -
ri
using the definition of r→ in terms of the three axes :
(✗ a
-
✗ , )I + (ya -
) j + (Zz 2) ti
y , A ✗ i + ☐ yj - =
+ ☐z ti
Vaverage Crate of change position )
velocity
=
:
Vaverage = 2¥ it 1¥ É Varg I + Varg
j
+ =
-
✗ -
yj + Varg -
z
É
Vinstantaneous = ¥-70 %¥= 8¥ ¥ it dd¥J + ¥ É = =
✓✗ I + Vyjtvz E
this is the vector sum of instantaneous velocities in
the ✗
, y ,
and 2- directions
acceleration vector derivation similar to velocity :
instantaneous acceleration :
a→= ¥18 F¥=8¥ (time derivative of the velocity vector)
A.instantaneous :
= 8¥ : +8¥ ; +8¥ E
= a I + ayjtazk
✗
vector sum of instantaneous accelerations in three directions
different way of looking at the acceleration vector
redefine velocity : ri ( magnitude ✗ direction)
✓ =
IF I =
vi. + Kit Vz
'
i = Tlv
a→=daY=a¥i +8¥
parallel acceleration : oi ,,
¥-0 =
normal to acceleration : at vd¥ =
applications :
projectiles (warfare and ball games)
circular motion (adventure parks)
projectile motion :
projectile objects :
that follow trajectory with given initial velocity determined by gravitation
(and air resistance)
'
a- (t) = 8¥ it -8¥ j -
projectile trajectory :
initial conditions E- 0 :
✗ =
0 , y = 0
Vox =
V0 COS do
Voy =
Vo sin do
y [✗ ) =
-
É9✗Y( V02 COST ao)) + tan (ao )
highest point ¥-0 Ymax Evo's in Tao )/g
=
:
distance of impact :
y =
0 ✗=0 or ✗ = R =
Voisin ( Zoro) 1g
motion in two or three dimension
define coordinate system ( define origin & the ✗ -
, y -
,
and z-axis directions )
In 3D, determine position ( r ) at any moment in time
any position vector can be written as the sum of three vectors in the direction of each axis
trajectory : the path an object travels over time Fct) =
✗ (t ) it yet )j + 2 A) it
this requires you to find functions of change in position relative to each single axis over
time ( ✗ (t ) , y (t ) , 2- (t ) )
motion along a straight line can be used for each axis separately (axes are independant )
displacement and average velocity
displacement F' Ñ (ta) Flt ) : = -
.
= ri -
ri
using the definition of r→ in terms of the three axes :
(✗ a
-
✗ , )I + (ya -
) j + (Zz 2) ti
y , A ✗ i + ☐ yj - =
+ ☐z ti
Vaverage Crate of change position )
velocity
=
:
Vaverage = 2¥ it 1¥ É Varg I + Varg
j
+ =
-
✗ -
yj + Varg -
z
É
Vinstantaneous = ¥-70 %¥= 8¥ ¥ it dd¥J + ¥ É = =
✓✗ I + Vyjtvz E
this is the vector sum of instantaneous velocities in
the ✗
, y ,
and 2- directions
acceleration vector derivation similar to velocity :
instantaneous acceleration :
a→= ¥18 F¥=8¥ (time derivative of the velocity vector)
A.instantaneous :
= 8¥ : +8¥ ; +8¥ E
= a I + ayjtazk
✗
vector sum of instantaneous accelerations in three directions
different way of looking at the acceleration vector
redefine velocity : ri ( magnitude ✗ direction)
✓ =
IF I =
vi. + Kit Vz
'
i = Tlv
a→=daY=a¥i +8¥
parallel acceleration : oi ,,
¥-0 =
normal to acceleration : at vd¥ =
applications :
projectiles (warfare and ball games)
circular motion (adventure parks)
projectile motion :
projectile objects :
that follow trajectory with given initial velocity determined by gravitation
(and air resistance)
'
a- (t) = 8¥ it -8¥ j -
projectile trajectory :
initial conditions E- 0 :
✗ =
0 , y = 0
Vox =
V0 COS do
Voy =
Vo sin do
y [✗ ) =
-
É9✗Y( V02 COST ao)) + tan (ao )
highest point ¥-0 Ymax Evo's in Tao )/g
=
:
distance of impact :
y =
0 ✗=0 or ✗ = R =
Voisin ( Zoro) 1g
