Slope of the tangent on a position-displacement curve represent
which physical quantiy ? Explain.
Answers
Answer:
Modern mathematical notation is a highly compact way to encode ideas. Equations can easily contain the information equivalent of several sentences. Galileo's description of an object moving with constant speed (perhaps the first application of mathematics to motion) required one definition, four axioms, and six theorems. All of these relationships can now be written in a single equation.
v = ∆s
∆t
When it comes to depth, nothing beats an equation.
Well, almost nothing. Think back to the previous section on the equations of motion. You should recall that the three (or four) equations presented in that section were only valid for motion with constant acceleration along a straight line. Since, as I rightly pointed out, "no object has ever traveled in a straight line with constant acceleration anywhere in the universe at any time" these equations are only approximately true, only once in a while.
Equations are great for describing idealized situations, but they don't always cut it. Sometimes you need a picture to show what's going on — a mathematical picture called a graph. Graphs are often the best way to convey descriptions of real world events in a compact form. Graphs of motion come in several types depending on which of the kinematic quantities (time, position, velocity, acceleration) are assigned to which axis.
position-time
graph
Let's begin by graphing some examples of motion at a constant velocity. Three different curves are included on the graph to the right, each with an initial position of zero. Note first that the graphs are all straight. (Any kind of line drawn on a graph is called a curve. Even a straight line is called a curve in mathematics.) This is to be expected given the linear nature of the appropriate equation. (The independent variable of a linear function is raised no higher than the first power.)
Compare the position-time equation for constant velocity with the classic slope-intercept equation taught in introductory algebra.
s = s0 + v∆t
y = a + bx
Thus velocity corresponds to slope and initial position to the intercept on the vertical axis (commonly thought of as the "y" axis). Since each of these graphs has its intercept at the origin, each of these objects had the same initial position. This graph could represent a race of some sort where the contestants were all lined up at the starting line (although, at these speeds it must have been a race between tortoises). If it were a race, then the contestants were already moving when the race began, since each curve has a non-zero slope at the start. Note that the initial position being zero does not necessarily imply that the initial velocity is also zero. The height of a curve tells you nothing about its slope.
On a position-time graph…
slope is velocity
the "y" intercept is the initial position
when two curves coincide, the two objects have the same position at that time
graph
In contrast to the previous examples, let's graph the position of an object with a constant, non-zero acceleration starting from rest at the origin. The primary difference between this curve and those on the previous graph is that this curve actually curves. The relation between position and time is quadratic when the acceleration is constant and therefore this curve is a parabola. (The variable of a quadratic function is raised no higher than the second power.)