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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Substituting t0 0 and t1 2u0sin0 g in equation 1 above

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Unformatted text preview: 3, . . . respectively. So over each interval [t0, t 1], [t1, t2], [t2, t 3] and so on, yi(t) = u i .sin θ i .t − 1 g t 2 for i = 0, 1, 2, 3, . . . respectively. -2 t0 46 46 Let us take the general equation y(t) = u.sinθ.t − 1 g t2 that describes the height . -2 of a projectile and adapt it to each interval [t 0, t1], [t 1, t2], [t2, t3]. Over [t 0, t1] the height function y(t) is : y0(t) = u0.sinθ0.(t − t0) − 1 g (t − t0)2 . -2 Over [t 1, t2] the height function y(t) is : y1(t) = u1.sinθ1.(t − t1) − 1 g (t − t1)2 . -2 Over [t 2, t3] the height function y(t) is : y2(t) = u2.sinθ2. (t − t2) − 1 g (t − t2)2 -2 and so on. Let us first look at the behaviour around t = t0. Approaching t0 from the left : y(t) is not defined. We do not know the position or left height of the ball. Hence y(t) is not continuous from the left. We say that y(t) is not continuous left discontinuous at t = t0. discontinuity We may remove this discontinuity by defining the height of the ball when at rest discontin on the ground to be zero. In this case y(t) = 0 for t ≤ t0. So, approaching t0 from the left y(t) = 0. left Now, approaching t0 from the right : y0(t) = u0. sinθ0.(t − t0) − 1 g (t − t0)2 . right -2 Near t0 and to the right of t0 : t = t0 + δt . So (t − t0) = δt . right L imit y0(t) = Limit y0(t0 + δt) = Limit { u0.sinθ0.(δt) − 1 g (δt)2 } = 0 -2 δt → 0 δt → 0 t → t0+ Hence, L imit y0(t) = 0 = L imit t → t0− t → t0+ y0(t) Value y0(t) = y0(t0) = 0 t = t0 Since, L imit − y0(t) = L imit + y0(t) = Value y0(t) = 0 t → t0 we say : t → t0 t = t0 y0(t) is continuous at t = t0. continuous 47 Let us now look at the behaviour around t = t1. Approaching t1 from the left y(t) is : y0(t) = u0.sinθ0. (t − t0) − 1 g (t − t0)2. left -2 Near t1 and to the left of t1 : t = t1 − δt . left So, y0(t) = u0.sinθ0.(t1 − δt − t0) − 1 g (t1 − δt − t0)2. -2 L imit y0(t) = Limit y0(t) = u0.sinθ0.(t1 − t0) − 1 g (t1 − t0)2 - - - (...
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