By Kleppner D., Kolenkow R.
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Additional info for An Introduction to Mechanics
We assume that we have a continuous record of position versus time. The average velocity v of the point between two times t1 and t2 is defined by x(t2 ) − x(t1 ) v= . ) The instantaneous velocity v is the limit of the average velocity as the time interval approaches zero: x(t + Δt) − x(t) . v = lim Δt→0 Δt The limit we introduced in defining v is exactly the definition of a derivative in calculus. In the latter half of the seventeenth century Isaac Newton invented calculus to give him the tools he needed to analyze change and motion, particularly planetary motion, one of his greatest achievements in physics.
Dr dt = A(αeαt ˆi − αe−αt ˆj) v= or v x = Aαeαt vy = −Aαe−αt . The magnitude of v is v x 2 + vy 2 √ = Aα e2αt + e−2αt . v= To sketch the trajectory it is often helpful to look at limiting cases. At t = 0, we have r(0) = A (ˆi + ˆj) v(0) = αA (ˆi − ˆj). Note that v(0) is perpendicular to r(0). y A v(0) r(0) r(t ) ˆj iˆ A v(t ) Trajectory v(t >> 0) x 18 VECTORS AND KINEMATICS As t → ∞, eαt → ∞ and e−αt → 0. In this limit r → Aeαt ˆi, which is a vector along the x axis, and v → αAeαt ˆi; in this unrealistic example, the point rushes along the x axis and the speed increases without limit.
The sketch shows a comparison of Δ f ≡ f (x + Δx) − f (x) with the linear extrapolation f (x)Δx. It is apparent that Δ f, the actual change in f (x) as x is changed, is generally not exactly equal to Δ f for finite Δx. We shall use the symbol dx, called the diﬀerential of x, to stand for Δx. The diﬀerential of x can be as large or small as we please. We define d f , the diﬀerential of f , by d f ≡ f (x)dx. x x x + Δx f (x) Δf df x x + dx x This notation is illustrated in the sketches. The symbols dx and Δx are used interchangeably but d f and Δ f are diﬀerent quantities: d f is a diﬀerential defined by d f = f (x)dx, whereas Δ f is the actual change f (x + dx) − f (x).