Summary Chapter 11- Rectilinear Motion

Vocabulary/Concepts List

force
mass
time
space
mechanics
particle
rigid body
statics
dynamics
kinematics
kinetics
translation
rotation
reference frame
Newtonian frame / absolute frame / inertial frame / fixed frame
rectilinear motion
position coordinate
velocity (average and instantaneous)
speed
acceleration (average and instantaneous)
uniform rectilinear motion
uniformly accelerated rectilinear motion
positive acceleration, negative acceleration
true acceleration, deceleration
relative motion
dependant motion
degrees of freedom of a system
curvilinear motion (2D & 3D)
position vector
velocity vector
acceleration vector
vector functions
derivatives of vector functions
projectile motion
motion curves
hodograph
x / y / z coordinate system
x / y /z components
osculating plane
tangential / normal / binormal coordinate system
tangential / normal / binormal components
radial / transverse / axial coordinate system
radial / transverse / axial components

Rectilinear Motion
In rectilinear motion the motion of a particle is along a straight path, or a line.
Hence, the position of the particle can be identified solely with a scalar value called the position coordinate, x.
This position coordinate, x, is the distance from a reference point chosen on the the line (i.e. the path of motion.)

The kinematic relationships for rectilinear motion:

v = dx/dt

a = dv/dt

a = d²x /dt²

a = v dv/dx

Note that these are all scalar variables and hence, scalar equations.
A positive or negative value of x indicates on which side of the reference point the body is at.
A positive or negative value of v indicates in which direction the particle is moving.
A positive or negative value of a each indicates one of two possible cases:

A positive value of a indicates that either

1) the particle is traveling in the positive direction AND it is increasing in speed
-or-
2) the particle traveling in the negative direction AND it is decreasing in speed

A negative value of a indicates that either

1) the particle is traveling in the positive direction AND it is decreasing in speed
-or-
2) the particle traveling in the negative direction AND it is increasing in speed

A Special Case of Rectilinear Motion:
Uniform Rectilinear Motion
Uniform rectilinear motion is when the acceleration is zero.

The kinematic relationships for uniform rectilinear motion:

a = 0

v = v_o = constant

x = x_o + v_ot

Again, note that these are all scalar variables and hence, scalar equations.

Another Special Case of Rectilinear Motion:
Uniformly Accelerated Rectilinear Motion
Uniformly accelerated rectilinear motion is when the acceleration is constant.

The kinematic relationships for uniformly accelerated rectilinear motion:

v = v_o + at

x = x_o + v_ot + ½ a t²

v² = v_o²+ 2 a (x - x_o)

Again, note that these are all scalar variables and hence, scalar equations.

Relative Rectilinear Motion

Consider two particles, A and B, moving in rectilinear motion.

x_B/A = x_B - x_A = the position of B relative to A (The vector goes from A to B)

v_B/A = v_B - v_A= the velocity of B relative to A

a_B/A = a_B - a_A= the acceleration of B relative to A

Curvilinear Motion
In curvilinear motion the motion of a particle is along a 2-dimensional or a 3-dimensional path, or curve.
Hence, the position of the particle must be identified with a 2-dimensional or a 3-dimensional vector called the position vector, r.
This position vector, r, is the displacement vector from the origin of a frame of reference.

The kinematic relationships for curvilinear motion:

r = ?

v = dr/dt

a = dv/dt

The question mark here refers to the fact that r can be represented using a variety of coordinate systems.
Since these are vector quantities, we need to know how to take the derivative of vector functions:

d(P + Q)/du = dP/du + dQ/du

d(fP) /du = df/du P + f dP/du

d(P · Q)/du = dP/du · Q + P · dQ/du

d(PxQ) = dP/du x Q + P x dQ/du

Projectile Motion

Projectile motion is the motion of a particle in which the only force acting on it is the earth’s gravitational field.

The motions in the three coordinate directions are independent and hence can be treated like URM or UARM.

The kinematic relationships for projectile motion:

x direction: URM	y direction: UARM	z direction: URM
a_x= 0	a_y = -g	a_z= 0
v_x = v_xo = constant	v_y= v_yo- gt	v_z = v_zo = constant
x = x_o + v_xot	y = y_o + v_yot - ½ g t²	z = z_o + v_zot

Relative Curvilinear Motion

Consider a particle A moving with respect to (wrt) a fixed frame Oxyz.

Consider a frame Ax’y’z’ attached to particle A and moving in translation ONLY wrt Oxyz.

The relative motion of a particle B moving wrt the frame Ax’y’z’ can be written as:

r_B/A = r_B - r_A = the position of B relative to a frame attached to A

v_B/A = v_B - v_A= the velocity of B relative to a frame attached to A

a_B/A = a_B - a_A= the acceleration of B relative to a frame attached to A

NOTE:

These equations are only valid if the frame Ax’y’z’ is moving in translation only wrt Oxyz.

In general the rate of change of a vector (v_B/A and a_B/A ) as observed from a moving frame is DIFFERENT from its rate of change as observed from a fixed frame. However, if the frame is moving in translation ONLY, the rate of change will be the same as observed from both frames.

Position, velocity and acceleration vectors represented by

rectangular components

r = xi + yj + zk

v = dx/dt i +dy/dt j + dz/dt k

a = d²x/dt² i + d²y/dt² j + d²z/dt² k

Where:

i is the unit vector in the x direction
j is the unit vector in the y direction
k is the unit vector in the z direction (k = i x j)

Note: These do not change in magnitude or direction as the motion proceeds.

Position, velocity and acceleration vectors represented by

normal and tangential components

v = v e_t

a = dv/dt e_t + v²/pho e_n

Where:

e_t is the unit vector in the tangential direction.
It is the direction of the particle’s velocity.
e_nis the unit vector in the (principle) normal direction.
It is in the osculating plane and directed towards the center of curvature of the path of motion.
e_bis the unit vector in the binormal direction.
It is the direction is perpendicular to the osculating plane. (e_b= e_t x e_n )
rho is the radius of curvature of the path at the point where the particle is at.

Note:

These components should be used when the path of motion is known.
These unit vectors change in magnitude and direction as the motion proceeds.
The normal and binormal components of velocity are always zero.
The binormal component of the acceleration is always zero.
a_t ( = dv/dt) indicates the change in speed
a_n ( = v²/rho ) indicates the change in direction

Position, velocity and acceleration vectors represented by

radial and transverse components

r = re_r+ zk

v = (dr/dt) e_r + (r dq/dt) e_theta+ (dz/dt) k

a = {d²r/dt²- r(dq/dt)²} e_r+ {r(d²q/dt²)+2(dr/dt)(dq/dt)} e_theta+ (d²z/dt²)k

Where:

e_r is the unit vector in the radial direction.
It is the direction the particle would move if r increased and theta remained constant.
e_thetais the unit vector in the transverse direction.
It is the direction the particle would move if theta increased and r remained constant.
k is the unit vector in the z direction. (k = e_r x e_theta)

Note:

These unit vectors change in magnitude and direction as the motion proceeds.
a_theta≠ dv_theta/dt
a_r≠ dv_r/dt