Unformatted text preview: m rest
(0 kmph) and accelerates steadily due East along the xaxis for 10 secs and
then levels off at 90 kmph (25 m/sec). Is v(t) continuous at t = 10 secs ?
continuous
v(t)
30
25 m/sec 25
20 15
S
P 10
E
E5
D
(0,0
0,0)
(0,0 ) v(t) = a . t for t ≤ 10 secs where
a is the constant acceleraton.
25 m/sec for t ≥ 10 secs. 5 10
15
TIME [sec] 20 25 t What acceleration are we going to use ? We still do not know how to differ entiate
differentia
dif entiate
the speed fuction v(t) to find the acceleration (rate of change of speed) at any
any
instant . Since the acceleration is steady or constant we may use the average
a verage
acceleration between any two instants , say t = 0 secs and t = 10 secs, where the
speeds are known.
a= v(t = 10 sec) − v(t = 0 sec)
25 m/sec
=
= 2.5 m/sec2
10 sec − 0 sec
10 sec Approaching t = 10 secs from the left : v(t) = a.t = 2.5 t meter/sec2
left
Near t = 10 secs and to the left of t = 10 secs :
left t = 10 − δt L imit − v(t) = Limit {a (10 − δt)} = Limit {10a −10δt} = 10a = 25 m/sec
a
a
δt → 0 a
δt → 0
t → 10
45 Approaching t = 10 secs from the right: v(t) = 25 meters/sec, constant.
right
Near t = 10 secs and to the right of t = 10 secs :
right L imit t → 1 0+ v(t) = Limit δt → 0 t = 10 + δt { 25 meter/sec } = 25 m/sec imit
imit
Hence, L→ − v(t) = 25 m/sec = tL→ 10+ v(t)
t 10 Value v(t) = v(t = 10 secs) = 25 meters / sec t = 10 imit
imit
Since, L→ 10− v(t) = tL→ 10+ v(t) = tValue v(t) = 25 m/sec
t
= 10
we say : v(t) is continuous at t = 10 secs.
continuous
Example 4: Is the function y(t), that describes the height of a bouncing ball,
continuous at instants t0, t1, t2, t3, . . . ?
y(t)
h
e
i
g
h
t a bouncing ball n t
t1
t3
t2
In general, y(t) = u.sinθ.t − 1 g t2 where u is the initial velocity and θ is the angle
2
of projection. In this example, over each interval [t0, t 1], [t1, t2], [t2, t3] and
so on, we have different initial velocities u0, u1, u2, u3, . . . and angles of projection
θ0, θ1, θ2, θ...
View
Full Document
 Fall '09
 TAMERDOğAN
 Limit, Δx

Click to edit the document details