Download Relevant Equations

Particle Rectilinear Motion
s = position
ds
dt
dv d 2 s
a

dt dt 2
ads  vdv
v
Constant Acceleration Equations
v  vo  ac t
1 2
ac t
2
v 2  vo2  2a c s  s o 
s  s o  vo t 
Projectile Motion
Horizontal Motion
v x  vo x
x  xo  vo x t
Vertical Motion
v y  vo  y  gt
y  y o  vo  y t 
v y2  vo  y
2
1 2
gt
2
 2 g  y  yo 
Particle Curvilinear Motion
x,y,z coordinates
v x  x
v y  y
v z  z
a x  x
a y  y
a z  z
n, t, z coordinates
v = velocity (always tangent to the path)
at = tangential acceleration
an = normal acceleration
ρ = radius of curvature
v  s
at  v  v
an 
dv
ds
v2

a  at2  a n2
3
  dy  2  2
1    
  dx  

d2y
dx 2
r, θ, z coordinates
vr = radial velocity
vθ = velocity in theta direction (perpendicular to ur vector)
v r  r
v  r
v z  z
a r  r  r 2
a  r  2r

a z  z
Relative Motion
vB = absolute velocity of object B
vA = absolute velocity of object A
vB/A = velocity of object B relative to object A
aB = absolute acceleration of object B
aA = absolute acceleration of object A
aB/A = acceleration of object B relative to object A
v B  va  v B / A
aB  a A  aB / A
Kinetics for System of Particles
W = weight (newtons or pound force)
Newtonian (x,y,z coordinates)
W  mg
Fx = sum of forces in the x-direction
Fy = sum of forces in the y-direction
Fz = sum of forces in the z-direction


 F  ma
 F  ma
 F  ma
x
x
y
y
Newtonian (n, t, z coordinates)
Ft = sum of forces in the tangential direction (to the path)
Fn = sum of forces in the normal direction (perpendicular to the tangent of the path)
F
F
F
t
 mat
n
 man
z
0
Newtonian (r, θ, z coordinates)
Fr = sum of forces in the radial direction
Fθ = sum of forces in the theta direction (perpendicular to the radial direction)
 F  ma
 F   ma
 F  ma
r
z
r
z
Work/Energy Theorem:
U1-2: Work of a non-conservative variable force
ΔT = change in kinetic energy
ΔVg = change in potential energy
ΔVe = change in potential energy (for a spring)
g = gravitational constant (9.81 meters per second squared or 32.2 feet per second squared)
h = height above or below reference datum (can be positive or negative)
k = spring constant
s = displacement from unstretched spring length
 
U 1 2   F dr

r2
r1
1 2
mv
2
V g   mgh
T
1 2
ks
2
U 12  T  Vg  Ve
Ve 
Conservation of Energy (no non-conservative forces):
T1  V g1  Ve1  T2  V g 2  Ve 2
Principle of Linear Impulse and Momentum
mv1 = Initial momentum
mv2 = final momentum



mv1    Fdt  mv 2
mv x1    Fx dt  mv x 2
mv y1    Fy dt  mv y 2
mv z1    Fz dt  mv z 2
Conservation of Linear Momentum for a system of particles
 m v   m v 
i
i 1
i
i 2
Central Impact
Coefficient of restitution, e: ratio of the restitution impulse to the deformation impulse
vB 2 = velocity of object B after impact
v A 2 = velocity of object A after impact
v B 1 = velocity of object B before impact
v A 1 = velocity of object A before impact
e
v B 2  v A 1
v A 1  v B 1
Oblique Impact
Along line of impact: use coefficient of restitution equation above
Along plane of impact: use conservation of linear momentum
Angular Impulse and Momentum for a system of particles:




 


r  mv1    r  F dt  r  mv 2
Conservation of Angular Momentum:

 

r  mv1  r  mv2