**Topic1-8:** Velocity vector and acceleration vector

The position of a particle in a two dimensional space can be described by
its x and y coordinates at each time t in a rectangular coordinate system.
Let us write the coordinates as x(t) and y(t). The speeds of the x-component,
v_{x}(t), and the y-component, v_{y}(t), are defined as

respectively. The velocity vector

The point (x(t), y(t)) will trace out a trajectory as time t varies.
Here we only consider a smooth trajectory that can be expressed as a
function F(x, y) = 0. Such a function can be considered as y = f(x), x = g(x),
or both. For example, the function y = A with A a given constant cannot be
expressed as x = g(y) but is in the form of y = f(x) = 0·x + A. Conversely,
the function

x = B can only be expressed as x = g(y) = 0·y + B. In the
remainder of this topic we only consider the case y = f(x). It is easy to carry
all the arguments to the case x = g(y) by interchanging x and y, and replacing
f by g everywhere.

Let us consider a trajectory y = f(x) where at time t the particle is at point P=(x(t), y(t)) on the trajectory. After a short time interval Δt, the particle moves to another point Q=(x(t+Δt), y(t+Δt)) on the trajectory. We can write

From the definition of differentiation, we know that

From the discussion of Topic 1-5 we also know that the left handside of the above equation is the slope of the contact line of the curve y = f(x) at point P. Since the limiting case Δx → 0 is induced by the limiting case Δt → 0, the above equation can be rewritten as

The velocity vector

There is another way to define the velocity vector **v**(t) from the
movement of the position vector of the particle itself. At time t the
particle is at the position P = (x(t), y(t)), and at time t+Δt it is at
point Q = (x(t+Δt), y(t_Δt)). The position vector OP is denoted as
**r**(t), and the position vector OQ as **r**(t+Δt) as shown in
Fig.1. A position vector that originates from point P and has a magnitude
equal to the length of PQ is denoted as Δ**r**. From the second kind
of addition of position vectors as discussed in Topic 1-7, we have

As Δt → 0, point Q moves toward point P along the trajectory, and the slope of the position vector Δ

The next step is to show that the velocity vector

The unit vector

The rectangular coordinate representation of the velocity vector

Furthermore, the aabove equations lead to

Eq.(7) shows that the magnitudes of

This can be rewritten as

From the sense of the direction of the movement of the particle, the + sign must be chosen, so we have u

To see how two velocity vectors can be added together, consider
a person walking on a moving platform. Let the person be at point P on the
platform at time t. The person is walking with a velocity vector **w** with
respect to the platform, whereas the platform, along with the point P, is
moving with a velocity **v** with respect to the ground as shown in part (a)
of Fig.3. We further denote that point P coincides with point Q of the ground
at time t. The velocity vector **v** forms an angle θ with a horizontal
line QT, and the velocity vector **w** forms and angle φ with QT as
shown in part (a) of Fig.3. At time t + Δt, point P of the platform
moves away from point Q of the ground, and the person walks away from point P
of the platform to point R of the platform as shown in part (b) of Fig.3. Let
the rectangular coordinate representations of points Q and R be
Q = (x, y) and

R = (x+Δx, y+Δy) respectively. The distance between
Q and P is |**v**|·Δt, and the distance between P and
R is |**w**|·Δt. Thus we have

Therefore, Eqs.(8) and (9) say

At the limit Δt → 0, we have

We can form a velocity vector

This

It should be noted that the addition of velocity vectors

An acceleration vector is defined according to its x and y components in the rectangular coordinate representation as

The the acceleration vector

To define the acceleration vector in a vector calculus fashion, we need to go into a velocity vector space with x-axis representing the x-components of velocity vectors and y-axis representing the y-components of velocity vectors. All the velocity vectors of the particle

In general if a motion (x(t), y(t)) is the superposition of two motions, such that

we can define two velocity vectors

Similarly two acceleration vectors

The overall velocity vector

respectively. From Eqs.(14), (15), (16) and (17), we have

Eqs.(18) imply that the formula of addition of vectors holds for both the velocity vectors and the acceleration vectors.

Here, we emphasize that both the velocity vector **v**(t) and the
acceleration vector **a**(t) of a particle in motion originate from the
point where the particle is at time t. The addition of two velocity vectors or
two acceleration vectors is defined similar to the addition of position vectors
of the first kind as discussed in Topic 1-7.

It is straightforward to extend the cases for two-dimensional velocity and acceleration vectors to three-dimensional spaces. The position of a moving particle in three dimensions can be written as

The velocity vector

and the acceleration vector

The addition of velocity vectors and acceleration vectors work the same as in the case of two-dimensional space. It should be noted that when two vectors originate from the same point, they always lie in a plane so that the graphic addition of vectors by drawing a parallelogram applies here too.

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