# Vector Essay

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…*

Example 1

The velocity and acceleration at time t

are: v(t) = r’(t) = 3t2 i + 2t j a(t) = r”(t) = 6t I + 2 j

VELOCITY & ACCELERATION

Example 1

The speed at t is:

2 2 2

| v(t ) |

(3t ) 9t

4

(2t )

2

4t

VELOCITY & ACCELERATION

Example 1

When t = 1, we have: v(1) = 3 i + 2 j a(1) = 6 i + 2 j |v(1)| = 13

VELOCITY & ACCELERATION

Example 1

These velocity and acceleration vectors are shown here.

VELOCITY & ACCELERATION

Example 2

Find the velocity, acceleration, and speed of a particle with position vector r(t) = ‹t2, et, tet›

VELOCITY & ACCELERATION

Example 2 t v(t ) r '(t )

2t , e , (1 t )e t t

a(t )

v '(t )

2

2, e , (2 t )e

2t 2

t

| v(t ) |

4t

e

(1 t ) e

2t

VELOCITY & ACCELERATION

The figure shows the path of the particle in

Example 2 with the velocity and acceleration vectors when t = 1.

VELOCITY & ACCELERATION

The vector integrals that were introduced in

Section 13.2 can be used to find position vectors when velocity or acceleration vectors

are known, as in the next example.

VELOCITY & ACCELERATION

Example 3

A moving particle starts at an initial position

r(0) = ‹1, 0, 0› with initial velocity

v(0) = i – j + k

Its acceleration is

a(t) = 4t i + 6t j + k

Find its velocity and position at time t.

VELOCITY & ACCELERATION

Example 3

Since a(t) = v’(t), we have: v(t) = ∫