Unizor - Physics - Mechanics - Kinematics - Uniform Motion

Notes to a video lecture on http://www.unizor.com



Uniform Motion



Let's consider a uniform motion from the position of Kinematics and
derive simple equations that mathematically describe this motion in
coordinate system.



The uniform motion along a straight line is characterized by two factors: its trajectory is a straight line in space and its velocity vector is constant, that is each component of the velocity vector does not change with time to assure the direction and magnitude of velocity are constant.

Actually, the second condition (velocity vector is constant) is sufficient for the first, which we will prove.



So, assume that velocity vector V is constant with coordinates {a, b, c}.

It means that, if x(t), y(t) and z(t) are coordinate functions defining the position vector P of our object, the derivatives of these functions are constant:

x'(t) = a

y'(t) = b

z'(t) = c

where a, b and c are constants
and at least one of these constants is not zero (otherwise, we deal
with a trivial case of object at rest considered in one of the earlier
lectures).

These three equations of motion for each coordinate can be written in vector form as

P' = V,

where derivative of a vector P' means a vector of derivatives {x'(t), y'(t), z'(t)} of each of its components.



The only function, whose derivative is constant, is a linear function.
Hence, the position of our object can be described by linear functions
of time:

x(t) = a·t + x₀

y(t) = b·t + y₀

z(t) = c·t + z₀

where x₀, y₀ and z₀ are some constants.

Obviously, the derivatives of these three functions representing the position correspondingly equal to a, b and c - components of our constant velocity vector of motion.

How to determine constants x₀, y₀ and z₀ that participate in the equations for position of our object?

Well, we cannot without additional information.

Notice that for t=0 (that is, at the beginning of motion)

x(0) = a·0 + x₀ = x₀,

y(0) = b·0 + y₀ = y₀,

z(0) = c·0 + z₀ = z₀.

Therefore, we can say that {x₀, y₀, z₀} is a point, where the motion starts. Knowledge of this initial position is needed to recreate the trajectory of an object moving uniformly in addition to the knowledge about its velocity vector.

Let vector P₀ represent the initial position {x₀, y₀, z₀}. Then the equations of motion can be expressed in vector form as

P(t) = t·V + P₀



By itself, velocity does not define the motion, we need an initial position as well.

Knowing both, an initial position and a constant vector of velocity, we can find equations of motion for x(t), y(t) and z(t).



Let's now prove that the motion described by equations

x(t) = a·t + x₀

y(t) = b·t + y₀

z(t) = c·t + z₀

goes along a straight line.



To prove this, we will prove that any two vectors from the point where the motion started P₀{x₀, y₀, z₀} to any two points along its trajectory P₁{x(t₁), y(t₁), z(t₁)} and P₂{x(t₂), y(t₂), z(t₂)} are collinear.

In other words, we will prove that for some real number k the following is true:

P₀P₁ = k·P₀P₂



Vector P₀P₁ has coordinates equal to a difference between coordinates of P₁{x(t₁), y(t₁), z(t₁)} and P₀{x₀, y₀, z₀}, that is

P₀P₁ =

={x(t₁)−x₀, y(t₁)−y₀, z(t₁)−z₀}=

= {a·t₁, b·t₁, c·t₁}



Vector P₀P₂ has coordinates equal to a difference between coordinates of P₂{x(t₂), y(t₂), z(t₂)} and P₀{x₀, y₀, z₀}, that is

P₀P₂ =

={x(t₂)−x₀, y(t₂)−y₀, z(t₂)−z₀}=

= {a·t₂, b·t₂, c·t₂}



Obviously, vectors

P₀P₁ = {a·t₁, b·t₁, c·t₁} =

= t₁·{a, b, c} = t₁·V

and

P₀P₂ = {a·t₂, b·t₂, c·t₂} =

= t₂·{a, b, c} = t₂·V

are collinear to each other:

P₀P₁ = (t₁/t₂)·P₀P₂



In addition, both these vectors are collinear to velocity vector V = {a, b, c}, which proves that the direction of a trajectory in uniform motion, defined by vector P₀P₁, is the same as the direction of the velocity vector {a, b, c}. So, an object in uniform motion always moves in a direction of the constant velocity vector.



End of proof.



Since uniform motion is a motion along a straight line, it's very
convenient to choose a system of coordinates, where X-axis coincides
with this line and the positive direction of the X-axis coincides with
the direction of the motion.

Then y(t)=z(t)=0 and y'(t)=z'(t)=0 for all moments of time t. In particular, velocity vector looks now as {a, 0, 0}.

For these cases traditionally the letter v is used for the X-component of the velocity vector.

The only non-trivial equation of motion then will be about x(t):

x(t) = v·t + x₀

If the origin of the system of coordinates coincides with the initial position of the object, x₀=0 and the equation of motion looks really simple:

x(t) = v·t



Since the positive direction of the X-axis is the same as the direction of the movement, velocity vector's direction plays minimal role, and only its magnitude (positive) is important, in which case we can use the term speed for v in the above equation of motion.





CONCLUSION

An object moving from the initial position P₀ = {x₀, y₀, z₀} with constant velocity vector V = {a, b, c} moves along a straight line collinear to a velocity vector and has the following equations of motion:

x(t) = a·t + x₀

y(t) = b·t + y₀

z(t) = c·t + z₀

Or in vector form

P(t) = t·V + P₀

If the X-axis coincides with the line of motion and a vector of velocity has X-component v (Y- and Z-comonents are, of course, equal to zero), the equations are simplified:

x(t) = v·t + x₀

y(t) = 0

z(t) = 0

If, in addition, the origin of coordinates coincides with the point of beginning of motion, that is x₀=0, the equations are simplified even further:

x(t) = v·t

y(t) = 0

z(t) = 0