Unizor - Physics4Teens - Mechanics - Kinematics - Speed, Velocity

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



Speed and Velocity



This is the first Physics lecture, where a solid mathematical background will be needed.

In particular, we assume the familiarity with vectors and derivatives.

These and many other mathematical concepts can be found in the course
"Math 4 Teens" on the same site as this lecture - UNIZOR.COM.



In the first approximation speed is a distance covered by a moving object in a unit of time.

What's wrong with this definition?

Firstly, a moving object moves with a different speed at different
moments in time, so speed is a function of time, which is not reflected
in this definition.

Secondly, what is a "unit of time"? Any unit of time, like a second, is
not a "moment" in time, but an interval, during which an object can
still change its speed.



Let's approach this concept more rigorously.

Our task is to define a speed as a characteristic of motion at any moment in time.



The simplest motion we consider first is a movement of an object along
the X-axis, according to the following function of X-coordinate of time t: x(t) = a·t - a uniform movement along the straight line (X-axis).



Let's define the average speed during any time interval from moment t₁ to moment t₂ as the distance covered during this time interval divided by the length of time this distance was covered.

For an object moving along the X-axis this average speed can be expressed in a formula:

s_ave(t₁, t₂) =

= [x(t₂) − x(t₁)] / (t₂ − t₁)



Remarkably, the average speed for an object uniformly moving along the X-axis as x(t) = a·t will be constant, independently of a choice of t₁ and t₂ - the beginning and ending moments in time:

s_ave(t₁, t₂) =

= [a·t₂ − a·t₁] / (t₂ − t₁) = a

Of course, this type of movement is called uniform exactly for this reason - average speed of movement is constant on any time interval.



Consider now a movement along the X-axis according to general function x(t) and the task of defining its speed at any moment t.

Let's note the position of our object at two moments in time - t and t+Δt. We can find an average speed during this time period:

s_ave(t, t+Δt) =

= [x(t+Δt) − x(t)] / (t+Δt − t) =

= [x(t+Δt) − x(t)] / Δt



It is reasonable now to define a speed at moment t, called instantaneous speed s(t), as a limit of the above expression for the average speed on interval [t, t+Δt], when Δt is diminishing to 0, that is time increment Δt is an infinitesimal variable:

s(t) =

= lim_Δt→0[x(t+Δt) − x(t)] / Δt



The limit above is a definition of the derivative of a function x(t). Therefore, the rigorous definition of an instantaneous speed of an object moving along the X-axis according to function of time x(t) is the derivative of this function:

s(t) = dx(t)/dt = x'(t)



In one of the previous lectures we mentioned that coordinate functions x(t), y(t) and z(t) must be continuous to prevent instant science fiction jumps to other planets. From now on we will also assume that these coordinate functions are differentiable to be able to determine the speed of movement as a derivative of a coordinate function by time.



This completes the definition of speed for a one-dimensional movement along the X-axis.

Now we will address general case of movement in three-dimensional space described by coordinate functions x(t), y(t) and z(t).



To define a speed in case of general three-dimensional movement represented by three coordinate functions x(t), y(t) and z(t), we will use the vector representation of the position of an object.



So, assume that at time from t to t+Δt a vector a from the origin of coordinates to point {x(t), y(t), z(t)} is transformed into vector b from the origin of coordinates to point {x(t+Δt), y(t+Δt), z(t+Δt)}.
Then the difference between these vectors is a displacement that
happened with a moving object during this time. The first important
thing to notice here is that this displacement is a vector d = b − a.



Though we did not mention the vector character of the movement along the
X-axis above, it was implicitly there, because movements forward
(towards increasing X-coordinates) or backward (towards decreasing
X-coordinates) are two different directions that are characteristic of
vectors. So, the positive or negative displacement [x(t+Δt) − x(t)] is really a vector because it has a sign - positive if x(t+Δt) > x(t) and negative if x(t+Δt) < x(t), and a sign in one-dimentional case of movement along the X-axis is the direction.



Now, when we have determined the vector character of a displacement from a position a at time t to position b at time t+Δt, we can define the average displacement during this period of time - it's a vector (b − a) / Δt.

Coordinates of this vector are, obviously, {Δx(t)/Δt, Δy(t)/Δt, Δz(t)/Δt}, where

Δx(t) = x(t+Δt) − x(t),

Δy(t) = y(t+Δt) − y(t),

Δz(t) = z(t+Δt) − z(t).



When time increment Δt is infinitesimal, all three components of the displacement vector are infinitesimal as well.

If all three coordinate functions are differentiable (which we will always assume), the limit of the average displacement vector (b − a) / Δt on a time interval Δt, when Δt→0, is a vector with coordinates {x'(t), y'(t), z'(t)}.



Hence, the instantaneous speed of movement of an object defined by coordinate functions x(t), y(t) and z(t) is a vector with coordinates {x'(t), y'(t), z'(t)}.

This vector is called velocity, while the term speed is reserved only for the magnitude of this vector, that is its absolute value, disregarding the direction, which is equal to

√[x'(t)]²+[y'(t)]²+[z'(t)]²