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)...
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- Fall '09
- TAMERDOğAN
- Limit, Δx
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