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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# and angles of projection 0 1 2 3 respectively

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Unformatted text preview: m rest (0 kmph) and accelerates steadily due East along the x-axis 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...
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