Curvilinear Motion: Acceleration
 Average acceleration : a = v/ t
 By connecting points drawn at tips of
arrowheads of v(t), a curve can be formed,
known as hodograph.
 Acceleration related to this curve in the
same manner as velocity to path.
 Instantaneous acceleration: a = dv/dt
 Direction is tangent to hodograph.
 NOT tangent to path (most cases).
 Typically points towards “inside” curve of
path.
Rectangular Components
 Previous definitions of position,
velocity, acceleration are co-ordinate
system independent.
 Equations can be written for position
vector r in terms of Cartesian coordinates.

Resolution of the vector into x, y, z
components.
 r(t)


= x(t) i + y(t) j + z(t) k
Magnitude of r: r2 = x2 + y2 + z2
Direction of r is unit vector: ur = r / r
Rectangular Components: Velocity/Accel.
 Using r(t) = x(t) i + y(t) j + z(t) k, and v= dr/dt,
 v = d(x i)/dt + d(y j)/dt + d(z k)/dt
 Since i, j, k are constant with respect to time:
v
dx ˆ dy ˆ dz ˆ
i
j
k
dt
dt
dt
x iˆ y ˆj z kˆ
 Can also be written as vx i + vy j + vz k
 Magnitude and direction found same way as position.
 Acceleration vector is similar.
Chain Rule
 The function defining the path of
the particle is generally a function
of co-ordinates.
 Motion of a particle along that path
is expressed as function of time.
 How is it possible to get from coordinate definition to time?

Recall chain rule: dy
dt
dy dx
dx dt
Concept Question
 The path of a particle is defined by y = 0.5x2. If
the component of its velocity along the x-axis at x
= 2 m is vx = 1 m/s, its velocity component along
the y-axis at this position is

A) 0.25 m/s
B) 0.5 m/s

C) 1 m/s
D) 2 m/s