Download EGR280_Mechanics_8_ParticleKinematics

EGR 280
Mechanics
8 – Particle Kinematics I
Dynamics
Two distinct parts:
• Kinematics – concerned with the mathematics that describe
the geometry of motion, without being concerned with why
that motion takes place.
• Kinetics – the study of the relationships between the forces
that act on a body and its resulting motion.
Rectilinear motion of particles (motion along a straight line)
Keep track of the position of the particle with a position coordinate x:
O
x
The time rate of change of the position is velocity:
v = dx/dt
The time rate of change of the velocity is acceleration:
a = dv/dt
and
a = dv/dt = d(dx/dt)/dt = d2x/dt2; a = d2x/dt2
a = dv/dt = (dv/dx)(dx/dt) = v(dv/dx); a = v(dv/dx)
3 classes of problems:
1. Acceleration a is given as a function of time:
integrate: v = ∫a dt; x = ∫v dt
2. Acceleration is given as a function of position:
integrate: ∫v dv = ∫a dx
3. Acceleration is given as a function of velocity:
integrate: ∫dt = ∫(1/a)dv or ∫dx = ∫(v/a)dv
If we know the acceleration not as an analytical function, but as discrete values in time,
we can still integrate these discrete values to find the velocity, and subsequently
integrate the velocity to find the position. Consider the function below:
f(t)
fi+1
fi
Δt
t
The integral of f(x) can be piecewise-linearly approximated as:

b
a
b
f (t )  
a
t
( f i  f i 1 )
2
This is known as the Trapezoidal Rule, and has local error of order (Δt)2, so the smaller
Δt is, the more accurately the numerical integration be performed.
Uniform rectilinear motion – when acceleration is zero
a = 0 = dv/dt
v0 = dx/dt
x = x0 + v0t
velocity is constant

t
0
x
vo dt   dx
x0
Uniformly accelerated rectilinear motion –when acceleration is constant
t
 a dt  
a0 = dv/dt
v = v0 + a0t
v = dx/dt
x = x0 + v0t + ½ a0t2
a0 = v(dv/dx)
v2 = v02 + 2a0(x-x0)
0

t
0
0
v
v0
dv
x
(v0  a0 )dt   dx
x0

x
x0
v
a0 dx   vdv
v0
Motion of several particles
0
A
xB
xB/A is the position of B with respect to A
xB = xA + xB/A
Taking time derivatives:
vB = vA + vB/A
aB = aA + aB/A
xAB = xB/A
xA
xAB is the position from A to B
xB = xA + xAB
vB = vA + vAB
aB = aA + aAB
B