Chapter 13 – Vector Functions
13.4 Motion in Space: Velocity and Acceleration
Objectives:
 Determine how to calculate
velocity and acceleration.
 Determine the motion of an
object using the Tangent
and Normal vectors.
13.4 Motion in Space: Velocity and Acceleration
1
Position Vector



Suppose a particle moves through space so that its position
vector at time t is r(t).
Notice from the figure that, for small values
of h, the vector
approximates the direction of
the particle moving along
the curve r(t).
Its magnitude measures the
size of the displacement vector
per unit time.
13.4 Motion in Space: Velocity and
Acceleration
2
Velocity Vector

The vector 1 gives the average velocity over a
time interval of length h and its limit is the
velocity vector v(t) at time t :

The velocity vector is also the tangent vector and
points in the direction of the tangent line.
13.4 Motion in Space: Velocity and
Acceleration
3
Speed

The speed of the particle at time t is the
magnitude of the velocity vector, that is, |v(t)|.
| v ( t ) | | r '( t ) |
ds
= rate of change w .r.t. tim e
dt

For one dimensional motion, the acceleration of
the particle is defined as the derivative of the
velocity:
a(t) = v’(t) = r”(t)
13.4 Motion in Space: Velocity and
Acceleration
4
Visualization

Velocity and Acceleration Vectors
13.4 Motion in Space: Velocity and
Acceleration
5
Example 1

Find the velocity, acceleration, and
speed of a particle with the given
position function.
r ( t )  t i  ln t j  t k
2
13.4 Motion in Space: Velocity and
Acceleration
6
Newton’s Second Law of Motion
If the force that acts on a particle is
known, then the acceleration can be found
from Newton’s Second Law of Motion.
 The vector version of this law states that
if, any any time t, a force F(t) acts on an
object of mass m producing an
acceleration a(t), then

F(t) = ma(t)
13.4 Motion in Space: Velocity and
Acceleration
7
Example 2 – pg. 871 # 28

A batter hits a baseball 3 ft above the
ground toward the center field fence,
which is 10 ft high and 400 ft from
home plate. The ball leaves the bat
with speed 115 ft/s at an angle of 50o
above the horizontal. Is it a home
run? (Does the ball clear the fence?)
13.4 Motion in Space: Velocity and
Acceleration
8
Tangential and Normal
Components of Acceleration

When we study the motion of a particle, it
is often useful to resolve the acceleration
into two components:
◦ Tangential (in the direction of the
tangent)
◦ Normal (in the direction of the normal)
a  v 'T   v N
2
13.4 Motion in Space: Velocity and
Acceleration
9
Tangential and Normal
Components of Acceleration

Writing aT and aN for the tangential and normal
components of acceleration, we have
a = aTT + aNN
where
aT = v’
and
aN = v2
13.4 Motion in Space: Velocity and
Acceleration
10
Tangential and Normal
Components of Acceleration

We will need to have aT = v’ and
aN = v2 in terms of r, r’, and r”. To
obtain these formulas below, we start
with v · a.
aT  v ' 
r '( t ) r ''( t )
aN   v 
2
r '( t )
r '( t )  r ''( t )
r '( t )
13.4 Motion in Space: Velocity and
Acceleration
11
Example 3 – pg. 871 # 38

Find the tangential and normal
components of the acceleration
vector.
r ( t )  1  t  i   t  2 t  j
2
13.4 Motion in Space: Velocity and
Acceleration
12
Kepler’s Laws
Note: Read pages 844 – 846.
1. A planet revolves around the sun in an
elliptical orbit with the sun at one focus.
2.
The line joining the sun to a planet
sweeps out equal areas in equal times.
3.
The square of the period of revolution of
a planet is proportional to the cube of
the length of the major axis of orbit.
13.4 Motion in Space: Velocity and
Acceleration
13
More Examples
The video examples below are from
section 13.4 in your textbook. Please
watch them on your own time for
extra instruction. Each video is
about 2 minutes in length.
◦ Example 3
◦ Example 5
◦ Example 6
13.4 Motion in Space: Velocity and
Acceleration
14
Demonstrations
Feel free to explore these
demonstrations below.
Kinematics of a Moving Point
 Ballistic Trajectories

13.4 Motion in Space: Velocity and
Acceleration
15