Motion is a change of position, so to discuss motion, we must first discuss position. To discuss position, we must choose an origin and a reference direction. These choices are arbitrary and must be made before the position can be defined.
DISTANCE/ANGLE METHOD
As in one dimension, we may describe the position of an object by its distance from the origin and the direction in which it is displaced from the origin.
We choose a location for the origin and a reference direction. Tradi-tionally, the reference direction points to the right along a horizontal straight line.
We draw an arrow from the origin to the object. The length of the arrow is a distance and is called the magnitude of the position vector.
For us, a vector is a quantity that has both magnitude and direction.
The angle that the arrow line makes with the reference direction is taken as the direction of the position vector.
COMPONENT METHOD
For this description, we add a second reference direction. Calling the original direction (traditionally to the right) the x direction, our second direction is called the y direction.
The y direction is by definition perpendicular to the x direction. Tradi-tionally, this is taken to be upward on the page. When y points up (instead of down), with x to the right, we say that we have a right handed coordinate system.
The component description of a vector tells how much of the vector is along the x direction and how much is along the y direction. The figures below show two different vectors and their components.
VECTOR ALGEBRA
Vector Addition and Subtraction—Vectors are added following rules that work for adding steps in a journey on foot.
A+B is determined by placing the tail of the
B vector on the tip of the
A vector. The resultant vector, A +B is the vector from the tail of
A to the tip of
B, as in the figure below.
The vector difference
A−B is as the sum of the vector
A, with the
“reverse” of the vector
B, called −
B. As in the figure, −
B points opposite to
B.
Using the component method as sketched above, the magnitude of the
A+B and the reference direction (x=axis) is given
by tan [ ]
The results for the difference between A B
and can be found by simply placing a – sign in front of each B
above.
Scalar Multiplication of a Vector—When a vector is multiplied by a scalar (a number with no direction), the magnitude (length) of the vector is multi-plied by that number, and the direction is unchanged.
Scalar (“dot”) Product of Two Vectors—The scalar product,
A B • =ABcos( )θ =[A Bx x] [+ A By y] is the product of the magnitudes times the cosine of the angle between the two vectors. This product can be positive or negative, depending on the sign of the cosine.
Vector (“cross”) Product of Two Vectors—The vector product of two vectors has magnitude / / sin( )
A×B = AB θ .
The vector product also has direction. It is perpendicular to the plane defined by the
AandB vectors. There are two directions that satisfy that condition. The ambiguity is resolved with a right-hand rule.
If
AandB lie in the page, and if
A must turn clockwise to become parallel to
B
, then the direction ofA×B is into the page.
Note that A× = − ×B B A.
Displacement—The change in position is calculated by subtracting the initial position vector from the final position vector.
The algebraic representation of this calculation uses ras the symbol for position and ∆r as the symbol for change in position. We write:
∆r =rFINAL −rINITIAL
The arrows above r emphasize the vector nature of the position by reminding us that it has a direction as well as a magnitude.
∆
r is called the displacement of the object.
Velocity—Velocity is defined as the rate of change of position. Average velocity vector, <v>, is defined as the change in position, ∆
r, divided by the time required to make the change, ∆t:
< > =
v r
t
∆
∆
The instantaneous velocity vector is the limit of the average velocity as the time interval (and thus also the change in position) approaches zero. It is the time derivative of the position.
Acceleration—Acceleration is defined as the rate of change of velocity.
Average acceleration, <a>, is defined as the change in velocity, v, divided by the time required to make the change, ∆t:
< > =a v t
∆
∆
Instantaneous acceleration is the limit of the average acceleration as the time interval (and thus also the change in velocity) approaches zero. It is the time derivative of the velocity.
Near the surface of the earth, it is found that all freely falling objects have
the same acceleration, a m
Freely falling means that no force except gravity acts on the object. In particular, we ignore wind resistance. When an object is launched into the air with some initial velocity, it is freely falling after launch, even though it might not be moving downward.
The motion of such an object near the surface of the earth is the simplest example of projectile motion.
THE EQUATIONS OF MOTION AS A FUNCTION OF TIME If we choose our x-axis to be horizontal and our y-axis to be vertical (the traditional choices) then the acceleration in the x direction is zero, while the acceleration in the y direction is −9 8. m2
sec .
In these coordinates, the projectile motion separates; the x motion is independent of the y motion (as long as the object is free falling).
The equation of motion for the y component of the motion is
y m compo-nent of the initial velocity.
(The x and y components of initial velocity are found from the magni-tude and direction of the initial velocity. One uses the same trigonometric method as was used to find the x and y components of a position vector.) The y component of the velocity can be obtained by differentiation:
v m
The x component of the motion is more simple, since there is no acceleration in the x direction:
x=v t0x +x0
and the x component of the velocity is constant:
vx =v0x
THE TRAJECTORY
The trajectory is a plot of the y component of the motion versus the x component of the motion. Each point on the trajectory represents the position at a particular time. It can be found by solving the x equation for t and replacing t in the y equation:
y m
In either case, the trajectory is a segment of a parabola, curving downward. We may ask, for example, for the x component of the position when the particle has returned to its initial height. The answer is often called the range of the projectile.