3233 WordsNov 9, 201413 Pages

VECTOR FUNCTIONS
VECTOR FUNCTIONS
Motion in Space: Velocity and Acceleration
In this section, we will learn about:
The motion of an object using tangent and normal vectors.
MOTION IN SPACE: VELOCITY AND ACCELERATION
Here, we show how the ideas of tangent and normal vectors and curvature can be
used in physics to study:
The motion of an object, including its velocity and acceleration, along a space curve.
VELOCITY AND ACCELERATION
In particular, we follow in the footsteps of
Newton by using these methods to derive
Kepler’s First Law of planetary motion.
VELOCITY
Suppose a particle moves through space so that its position vector at
time t is r(t).
VELOCITY
Vector 1
Notice from the figure*…show more content…*

So, C=i–j+ k
VELOCITY & ACCELERATION
Example 3
It follows: v(t) = 2t2 i + 3t2 j + t k + i – j + k
= (2t2 + 1) i + (3t2 – 1) j + (t + 1) k
VELOCITY & ACCELERATION
Example 3
Since v(t) = r’(t), we have: r(t) = ∫ v(t) dt
= ∫ [(2t2 + 1) i + (3t2 – 1) j + (t + 1) k] dt = (⅔t3 + t) i + (t3 – t) j + (½t2 + t) k + D
VELOCITY & ACCELERATION
Example 3
Putting t = 0, we find that D = r(0) = i.
So, the position at time t is given by: r(t) = (⅔t3 + t + 1) i + (t3 – t) j + (½t2 + t) k
VELOCITY & ACCELERATION
The expression for r(t) that we obtained
in Example 3 was used to plot the path of the particle here for 0 ≤ t ≤ 3.
VELOCITY & ACCELERATION
In general, vector integrals allow us to recover:
Velocity, when acceleration is known
v(t )
v(t0 )
t t0
a(u ) du
Position, when velocity is known
r (t ) r (t0 )
t t0
v(u ) du
VELOCITY & ACCELERATION
If the force that acts on a particle is known, then the acceleration can be found from
Newton’s Second Law of Motion.
VELOCITY & ACCELERATION
The vector version of this law states that if,
at any time t, a force F(t) acts on an object of mass m producing an acceleration a(t),
then
F(t) = ma(t)
VELOCITY & ACCELERATION
Example 4
An object with mass m that moves in
a circular path with

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