02Classical Mechanics
A body in uniform motion covers equal displacements in equal intervals of time: its velocity never changes. It is the simplest variant of the invariant, the one obtained with \(a = 0\), and it is enough to describe a tortoise heading for the finish line, the sound of an echo or two cars on a highway. Here you can solve any stretch of uniform motion, average several stretches and find, by equations and graphically, when two bodies meet.
In uniform motion \(x(t)\) is a straight line and \(v(t)\) a horizontal one; the area of the rectangle under \(v(t)\) is the displacement.
With no acceleration the five equations of the invariant reduce to a single relation between displacement, velocity and time, plus the definition of displacement. Two data fix the third quantity.
Instants that follow from the motion fixed by the data.
When a trip is made of stretches at different constant speeds, the average velocity is the total displacement over the total time, not the mean of the velocities. Give each stretch its velocity and its length or duration.
Each body follows its own straight line \(x(t)\); where the lines cross, they meet. A body can start late, stop for a while (a stretch with \(v = 0\)) and set off again. With a finish line the module also decides the race and, if a body takes a nap, how long the nap can last for it to tie.
To describe a motion we treat the body as a particle: a point with mass and no size, so that all its parts move in exactly the same way. A train, a tortoise or an electron are particles as long as we ignore their rotations and vibrations. In one dimension its state is given by a single coordinate \(x\), measured from a chosen origin along an axis with a chosen positive direction.
The displacement between two instants is the change of position, \(\Delta x = x_f - x_i\); it has a sign and it ignores what happened in between. The distance travelled is the length of the path actually covered, always positive. Someone who walks 100 m and comes back has a displacement of zero and a distance of 200 m: the average velocity of a car over a full lap of a racing circuit is zero.
The average velocity over an interval is the displacement divided by the elapsed time,
\[\bar v = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1},\]the slope of the straight line that joins the two points of the graph \(x(t)\). The average speed is the distance divided by the time; it is never negative and it is at least the magnitude of the average velocity.
When the velocity is constant, the average velocity of every interval is the same number \(v\), and
\[x(t) = x_0 + v\,t.\]The graph \(x(t)\) is a straight line of slope \(v\) (positive if it moves towards increasing \(x\)), and \(v(t)\) is a horizontal line. Equal times give equal displacements, which is what the equally spaced dots of the motion diagram show.
The course obtains it by applying the conditions of uniform motion to the invariant \(V(t) = V_o + at\), \(V^2 = V_o^2 + 2aS\), \(S = V_o t + \tfrac{at^2}{2}\). With \(a = 0\):
\[V(t) = V_o,\qquad S(t) = V_o\,t,\qquad V(t)^2 = V_o^2.\]Nothing new has to be memorised: the general equations already contain this case.
Two bodies meet when they are at the same place at the same time. Writing the law of each one, \(x_A(t)\) and \(x_B(t)\), the meeting is the solution of \(x_A(t) = x_B(t)\). Graphically it is the crossing of the two lines in the plane \(t\)–\(x\); if one body starts late or stops for a while, its line is broken into segments and each segment is studied separately.
A cyclist who climbs at 5 km/h and descends at 20 km/h does not average 12.5 km/h unless both stretches last the same time. With equal lengths \(L\) the times are \(L/5\) and \(L/20\) and
\[\bar v = \frac{2L}{L/5 + L/20} = 8\ \text{km/h},\]the harmonic mean: the slow stretch lasts longer and weighs more in the average.
Problems adapted from Serway (chapters 2, 4, 5 and 6), Resnick and the course notes. Load any of them into the laboratory: the data fill the panel and the solver does the rest, so you can then change any value and see how the answer reacts.
A tortoise moves steadily at its maximum speed of \(0.200\ \text{m/s}\) towards a finish line \(1.00\ \text{km}\) away. How long does it take?
A car keeps a constant \(90\ \text{km/h}\) for \(15\ \text{min}\). What distance does it cover?
Sound travels in air at \(343\ \text{m/s}\). The echo of a shout comes back after \(2.4\ \text{s}\). How far did the sound travel, and how far is the wall?
A body at \(x_0 = 50\ \text{m}\) moves with a constant velocity of \(-4.00\ \text{m/s}\). Where is it after \(10.0\ \text{s}\)? When does it pass through the origin?
A particle in uniform motion goes from \(x = -20\ \text{m}\) to \(x = 60\ \text{m}\) in \(16\ \text{s}\). What is its velocity?
Theory after R. Resnick, D. Halliday and K. Krane, Física, vol. 1, chapter 2 (sections 2-1 to 2-4); problems from R. Serway and J. Jewett, Física para ciencias e ingeniería, vol. 1, chapters 2 and 4, and from the course notes and workshop.