Deriving an equation of motion using Calculus

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Milouw
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How can I derive this equation of motion (s=1/2(u+v) t) using calculus? Please I need this for my Math IA
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ghostwalker
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(Original post by Milouw)
How can I derive this equation of motion (s=1/2(u+v) t) using calculus? Please I need this for my Math IA
All suvat equations refer to motion with constant acceleration.

So, starting from dv/dt = a (a being the constant acceleration), we can integrate to find v, and integrate again to get s.

Then a substitution from the v we found will derive the desired formula.

Have a go.

PS: I'm not familiar with the syllabus, so this may not be the desired method.
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Milouw
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(Original post by ghostwalker)
All suvat equations refer to motion with constant acceleration.

So, starting from dv/dt = a (a being the constant acceleration), we can integrate to find v, and integrate again to get s.

Then a substitution from the v we found will derive the desired formula.

Have a go.

PS: I'm not familiar with the syllabus, so this may not be the desired method.
Could you show me step by step please? Im not too bright I apologize ;-;
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ghostwalker
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(Original post by Milouw)
Could you show me step by step please? Im not too bright I apologize ;-;
I'd really like to see you have a go yourself, to start. It is the sort of thing that should be in your textbook. Giving full solutions straight off isn't particularly helpful, and is against forum rules.
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Milouw
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(Original post by ghostwalker)
I'd really like to see you have a go yourself, to start. It is the sort of thing that should be in your textbook. Giving full solutions straight off isn't particularly helpful, and is against forum rules.
Its not in the Textbook. Its an extra research project that Im doing
I tried and did it this way

a=(v_2-v_1)/t
a=ⅆv/ⅆt
ⅆv=a(ⅆt)
s=∫vⅆt=∫(at+v)ⅆt
s=(at^2)/2+vt
s=((v_2-v_1)/t (t)^2)/2+vt
s=(v_2-v_1 )(t)/2+vt s
=((v_2-v_1 )t+2v_1 t)/2
s=(v_2 t-v_1 t+2v_1 t)/2
s=(v_2+v_1 )t/2

Does that make any sense?
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Vinny C
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You make it hard work:

Acceleration 'a' is constant (a requirement of Newton's laws of motion).

a = dv/dt (definition of acceleration) so:
dv/dt = a (a constant) so:
v = at + u (constant of integration)
v = ds/dt (definition of velocity) so:
ds/dt = at + u
s = 1/2 at² + ut
QED
Last edited by Vinny C; 3 years ago
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RogerOxon
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(Original post by Vinny C)
Acceleration 'a' is constant (a requirement of Newton's laws of motion).
Nope.
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Vinny C
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(Original post by RogerOxon)
Nope.
In the context of this question, yes.
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RogerOxon
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(Original post by Vinny C)
In the context of this question, yes.
Nope. SUVAT has nothing to do with Newton's laws of motion. Nothing.

Newton's laws do NOT require constant acceleration.
Last edited by RogerOxon; 3 years ago
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Vinny C
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#10
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(Original post by RogerOxon)
Nope. SUVAT has nothing to do with Newton's laws of motion. Nothing.
I know... but if it wasn't then you wouldn't end up with s = ut +1/2 at²

Give the guy a break...
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