D d we may drop the subscript of particular angle 1

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ar instant in time we get a numerical value. In Calculus we also learn how to interpret this value. We will be able to say things such as the height is increasing, decreasing, is at a maximum or a minimum and so on. If the duration of flight is T, then from the physics of the situation we know the maximum height is reached when t = T/ 2. We also know that at the maximum maximum height the vertical speed should be zero. Substituting t = T/ 2 in the vertical speed equation u . s i n θ -- g t = 0, we get T = 2u. sin θ / g . u. 74 ertical Example 4: What is the instantaneous ver tical speed of the bouncing ball instantaneous ver (i) over (t0, t1) ? (ii) over (t1, t2) ? (iii) at t1 ? Over [t0, t 1] the height y(t) is : y0(t) = u0.sinθ0.(t − t0) − 1 g (t − t0)2 . -2 Over [t1, t2] the height y(t) is : y1(t) = u1. sinθ1.(t − t1) − 1 g (t − t1)2 . -2 ver y(t) = v er tical position n t0 t1 t2 t3 t ver y ’(t) = ver tical speed ertical (i) Over (t0, t1) the instantaneous ver tical speed y0’(t) = u0.sinθ0 − g(t − t0). instantaneous ver (ii) Over (t1, t2) the instantaneous ver tical speed y1’(t) = u1.sinθ1 − g(t − t1). ertical instantaneous ver ertical (iii) At t1 we have two instantaneous ver tical speeds : the speed on impact at t1 instantaneous ver while descending and the speed after impact at t1 while rising . We know from descending rising the previous example the duration of flight over [t0, t1] is T0 = 2u0. sinθ/ g. To find the speed on impact at t1 while descending we substitute t0 = 0 and t = T0 descending in y0’(t) = u0.sinθ0 − g(t − t0) to get y0’(t− ) = − u0.sinθ0 . 1 After impact at t1 we know the speed on rising is y1’(t+ ) = +u1.sinθ1. Alternatively, rising 1 we may substitute t = t1 in y1’(t) = u1.sinθ1 − g(t − t1) to get the same result. 75 5: Example 5 Let us differentiate f(x) = sin (x) differentiate VERAGE step: 1. AVERAGE ste p: ∆ f(x) ∆x = f(x + ∆ x) − f(x) (x + ∆ x) − x = f(x + ∆ x) − f(x) ∆x = sin(x + ∆ x) − sin(x) ∆x = {sin(x) . cos(∆ x) + cos (x)...
View Full Document

Ask a homework question - tutors are online