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Introductory Physics Homework Help
Analyzing Motion: Deriving Displacement Graphs from First Principles
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[QUOTE="Old_sm0key, post: 6836987, member: 604198"] [B]Homework Statement:[/B] Not a homework question per se. I revisited introductory kinematics, as taught at British A Level, and am struggling to reproduce the stadnard plots for displacement and velocity versus time. [B]Relevant Equations:[/B] The standard 1D kinematics equations plus introductory differentiation Initial displacement is h above the ground ie ##s\left ( t =0\right )=h##. I've chosen the ground as the vertical origin with upwards as the positive direction. Gravity will therefore always act in negative direction throughout. [B]Here are the graphs I which to reproduce from first principles[/B], where the motion starts at the red vertical line owing to my initial conditions: [ATTACH type="full" width="223px" alt="kinematics.png"]319539[/ATTACH] [U]##0\leqslant t< A##[/U] Starting from ##v=u+at## we derive displacement thus: $$\int_{0}^{t} vdt^\prime=\int_{0}^{t}\left ( u+at \right )dt^\prime$$ $$s\left ( t \right )-h=ut+\frac{1}{2}at^2$$ Using conditions ##t=0,u=0, a=-g\, \forall t \to s\left ( t \right )=h-\frac{1}{2}gt^2## which would give a parabola for first portion of displacement graph. So happy. The corresponding velocity graph is ##\frac{\mathrm{d} s}{\mathrm{d} t}=-gt##, so giving a negative straight line. Happy. [U]##A\leqslant t< B##[/U] Now I'm stuck with the displacement curve because I cannot exploit ##t=0## on the integral limits to simplify the maths...? Plus it seems a fudge to suddenly invoke ##s\left ( t \right )=vt-\frac{1}{2}at^2## that I see in some textbooks. [/QUOTE]
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Analyzing Motion: Deriving Displacement Graphs from First Principles
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