Vector

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