Ultimate Vector Kinematics Assignment Definition
A quantity that has the same value when measured by different observers in different reference frames is called a scalar. Some of the quantities used in the previous chapter are scalars; for example, the displacement and the time interval .
Examples of physical quantities which are not scalar are the components of the position, velocity and acceleration along an axis. Changing the direction or origin of that axis, the values of these quantities also change.
It is useful to write the equations of physics so that they are equal in any reference frame and vectors can be used to achieve this goal. A typical example is the displacement vector, which is a directed line segment between two points P1 and P2 in space, where the first point is considered the initial point, or origin, of the segment and the second point is the terminal point of the segment.
For example, in Figure 2.1, represents the vector starting at P1 and ending at P2; the arrow in P2 indicates that it is the end point and an arrow is also used on top of the letter used to denote the vector, to make it clear that this is a vector and not a regular algebraic variable.
2.1.1. Vector properties
The distance between the initial and terminal points of a displacement vector is called its magnitude or length. In this book, if a vector is denoted by its magnitude is then denoted by (the same letter without an arrow on top of it). Since the distance between two points is a scalar, the magnitude of a vector is a scalar too. A vector is defined by its magnitude and its direction, which is the orientation of the line through the two points.
Two vectors are equal if, and only if, they have the same direction and magnitude regardless of their initial point. For example, in Figure 2.1 the vector between points P1 and P2 is equal to the vector between P3 and P4 and that is why they were both denoted by . The distance between P3 and P4 equals the distance between P1 and P2 and the lines going through those two pairs of points are parallel. Vector , from point P5 to P6, is not equal to because its magnitude and direction are different. This kind of vectors in which the initial point does not matter, are called free vectors.
In Figure 2.2, starting from the point P the displacement vector ends at point Q and then from that point the displacement vector leads to the final point R. Namely, the combined displacements and are equivalent to a displacement from point P to R, denoted in the figure by vector . The combined displacement defines the sum of vectors and
Therefore, the addition of two vectors is done by placing the initial point of the second vector on the terminal point of the first one and then joining the initial point of the first vector with the terminal point of the second one.
The equation + = implies that = − and Figure 2.2 shows that vector joins the terminal point of with the terminal point of , when those two vectors are placed on the same initial point. Thus, the subtraction of two vectors is done placing those two vectors on the same initial point and then joining the terminal point of the second vector to the terminal point of the first vector.
The addition of vectors is a commutative operation: placing the initial point of vector on the terminal point of vector leads to the same combined vector obtained by placing the initial point of vector on the terminal point of vector (see Figure 2.3). The sum of vectors and is the diagonal of the parallelogram formed by placing the vectors on a common initial point and then repeating each vector with its initial point on the terminal point of the other. The addition of several vectors also follows the associative law.
According to the definition of vector addition and subtraction, the sum of a vector with itself, + , is a vector with the same direction but magnitude twice as big. Subtracting a vector from itself, - , leads to the null vector (a vector whose initial and terminal points are the same). Generalizing these results, the product of a scalar and a vector , is defined as another vector in the same direction, if is positive, or in the opposite direction if is negative and with magnitude equal to . The product of a scalar and a vector is usually written with the scalar first, followed by the vector, but the order does not really matter. If is equal to zero, then is the null vector .
Any vector is equal to the product of its magnitude, , and a unit vector , with the same direction of but magnitude equal to 1 (see Figure 2.4). Thus, the unit vector defines the direction of the corresponding vector . In this book unit vectors are denoted by a letter with a hat, instead of an arrow, on top of it.
As mentioned in the previous chapter, the position of a point P in space can be given by three coordinates in some system of coordinates and the rectangular coordinates system was introduced. Figure 2.5 shows the rectangular coordinates ( , , ) of point P.
There are two different ways to define the directions of the three axes , e . The usual way to define those directions follows the right-hand rule: after closing the right hand fingers into the palm, the thumb, index and middle fingers are then opened making them form right angles among themselves; the direction of the axis will then be given by the index finger, the direction of the axis by the middle finger and the direction of the axis by the thumb. A rectangular coordinate system is defined by a point O which is the origin and three rectangular unit vectors, , and , which are perpendicular among themselves and define the directions of the 3 axes.
Any vector can be obtained from the addition of three vectors with directions parallel to the 3 axes,
where ( , , ) and ( , , ) are the rectangular components of the vectors. From the properties of the addition of vectors and the product of a scalar and a vector it follows that the sum of two vectors and can be obtained by adding their components respectively:
That is, the sum of two vectors is another vector with rectangular components equal to the sum of the original vectors components. Note that the direction and magnitude of a vector are independent of the coordinate system and origin used; however, its rectangular components ( , , ) are different for different rectangular coordinates systems. If two vectors are equal, then their rectangular components on the same system must also be equal.
The position vector of a point P, with coordinates ( , , ), is the vector with initial point on the origin and terminal point on P; it can be obtained adding 3 vectors along the three axis, as shown in Figure 2.5
Note that the components of the position vector are equal to the coordinates of the point P, ( , , ). The position vector depends on the choice of the origin of the system; therefore, it is not a free vector, since its initial point is always at the origin, and its magnitude and direction are not the same in coordinates systems with different origins.
2.1.2. Velocity and acceleration vectors
The trajectory of a moving point can be determined at every moment by an expression for the position vector of the point as a function of time
Each of the three components , and is a function of time. In a time interval the displacement of the point (see Figure 2.6) is equal to
where and are the position vectors at 1 and 2.
The vector obtained by dividing the displacement by is the average velocity, which has the same direction as the displacement . The velocity vector at a moment is defined as the limit of the average velocity, when the time interval , approaches zero, starting at time
Since the rectangular components of the displacement vector are , and , then the velocity vector is equal to
Applying equation 1.5 to each of the three components of the position vector, the three results can be combined into the following vector equation
The increase of the velocity vector, , in the time interval divided by that time interval defines the acceleration vector
and its rectangular components are the derivatives of the components of the velocity vector
The results obtained from equation 1.11 for each of the three components of the velocity vector can also be combined into a single vector equation
Equations 2.8 and 2.11 are the kinematic equations in 3 dimensions, written in vector form. Since the equality of two vectors implies equality of their components, those two equations are equivalent to
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