Callisto
Deriving kinematic equations
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Callisto
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For a = constant
Callisto
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\[v=\int a dt=at+c\]c = initial velocity
So,
\[v=u+at\]
gerryliyana
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good job
gerryliyana
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if a constan, a =0
v = u + at
v = u + 0
v = u
Callisto
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\[x_f = \int v(t) dt=\int (\int a dt) dt = \int(at+u)dt = \frac{1}{2}at^2 + ut +c\], c = initial position
So,
\[x_f = ut +\frac{1}{2}at^2 + x_i\]\[x_f-x_i = ut +\frac{1}{2}at^2\]\[s = ut+\frac{1}{2}at^2\]s=displacement.
gerryliyana
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for a = 0;
s = ut + 0.5 at^2
s = ut + 0
s = ut
Callisto
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From the third post,
\[s=ut+\frac{1}{2}at^2\]\[at^2+2ut -2s = 0\]\[t^2 + \frac{2u}{a}t - \frac{2s}{a}=0\]\[(t+\frac{u}{a})^2 - \frac{u^2}{a^2}-\frac{2s}{a}=0\]\[(t+\frac{u}{a})^2 = \frac{u^2-2as}{a^2}\]\[t+\frac{u}{a} = \frac{\sqrt{u^2-2as}}{a}\]\[t = \frac{\sqrt{u^2-2as}-u}{a} -(1)\]
Put (1) into v=u+at
\[v =u+a(\frac{\sqrt{u^2-2as}-u}{a})\]\[v = u +\sqrt{u^2-2as}-u\]\[v^2 = u^2 +2as\]
Callisto
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\[x_f = \int v(t) dt = v_{ave}t+C\]c = initial position
\[x_f = v_{ave}t +x_i\]\[x_f = \frac{1}{2}(u+v)t +x_i \]\[s = \frac{1}{2}(u+v)t\]displacement = s = \(x_f - x_i\)
Callisto
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Probably something wrong with the last post, which is s=(1/2) (u+v)t.
experimentX
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that's correct ... it assumes constant acceleration. that;s all.
Callisto
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Seriously?! I did it!?!
experimentX
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let me see how can i put it up logically.
experimentX
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|dw:1351833416633:dw|
experimentX
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now you just need to show that for constant accn V_av = (u+v)/2 ... wanna try it?
Callisto
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Huh!? I thought it's some maths.. Oh..How to start??
experimentX
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try using MVT
Callisto
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MVT.... again.../_\
experimentX
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well, you can do this without MVT ,,, i was wondering if i could improve my skills with MVT.
try expanding v(t) in the expression of average.
experimentX
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or, V_av = s/t
Callisto
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expanding v(t)?
experimentX
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v(t) = u + at
experimentX
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or simply put ,,, average velocity = distance/time
Callisto
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Fail ._.
Callisto
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\[v_f= v_i+\int_0^{t_f} a dt=v_i + at\]
So,
\[v_f=v_i + at\]
Callisto
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\[x_f = x_i+\int_0^{t_f} v(t) dt= x_i+\int_0^{t_f} (v_i+at) dt = x_i+v_it+\frac{1}{2}at^2\]
So,
\[x_f = x_i+v_it+\frac{1}{2}at^2\]
That is
\[x_f - x_i = v_it+\frac{1}{2}at^2\]\[s= ut+\frac{1}{2}at^2\]
experimentX
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|dw:1351853269560:dw|
experimentX
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|dw:1351853406639:dw|