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, vx(t), and the y-component, vy(t), are defined as


respectively. The velocity vector v(t) at time t is defined as a vector that originates from the point P = (x(t), y(t)) with x and y components of vx(t) and vy(t). Thus v(t) = (vx(t), vy(t)). The velocity vector v(t) is sometimes just called the velocity.

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 v(t) = (vx(t), vy(t)) lies on a line with a slope vy(t)/vx(t). Eq.(3) implies that the velocity vector v(t) originates from the point P and lies on the contact line of the trajectory at point P.

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 Δr. 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 Δr approaches that of the contact line of the trajectory at point P. The magnitude of Δr approaches 0 as Δt → 0. On the contact line APB as shown in Fig.2 a unit length vector k is defined such that it originates from point P, lies on the contact line APB, and points along the direction of the movement of the particle. A velocity vector u(t) is then defined as
        
The next step is to show that the velocity vector u(t) as defined in Eq.(4) is exactly the same as the velocity vector v(t) defined in Eq.(1). Let the angle between the contact line APB and a horizontal line passing through point P be denoted as θ as shown in Fig.2. The slope of the contact line APB is clearly tanθ. Eqs.(1) and (3) then lead us to
        
The unit vector k can be expressed in the rectangular coordinate representation as

The rectangular coordinate representation of the velocity vector u(t) is

Furthermore, the aabove equations lead to
        
Eq.(7) shows that the magnitudes of u(t) and v(t) are the same. Combining Eqs. (6) and (7), we have
        
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 ux(t) = vx(t). Eq.(6) then says uy(t) = vy(t). Therefore,
        

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


v and w can be written as v = (vx, vy) and w = (wx, wy) respectively. We know that

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

At the limit Δt → 0, we have

We can form a velocity vector U = (Ux, Uy) such that

This U is the velocity vector of the person viewed from the ground at time t when the person is at point P of the platform; of course point P coincides with the ground point Q at that moment as mentioned before. Eq.(10) then says that

It should be noted that the addition of velocity vectors v and w as described in Eq.(11) resembles the addition of position vectors of the first kind, that is, the two vectors to be added need to originate from the same point as discussed in Topic 1-7. Also the addition by drawing a parallelogram applies here as shown in Fig.4.

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


The the acceleration vector a(t) is defined as a vector originates from point P and with the rectangular coordinate representation as
        
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 v(t) are plotted as if originate from the origin of this velocity vector space. However, this kind of exercise, though will lead to the same results as in Eqs.(12) and (13), does not illuminate any more physical insight, and we will not go further into that direction.

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


we can define two velocity vectors v1(t) and v2(t) as

Similarly two acceleration vectors a1(t) and a2(t) are naturally defined as

The overall velocity vector v(t) and acceleration vector a(t) are defined as

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 v(t) = (vx(t), vy(t), vz(t)) is defined as
        
and the acceleration vector a(t) = (ax(t), ay(t), az(t)) is defined as
        
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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