 #1
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Summary:

I am studying Physics on my own. I understand the Kinematic equations and how to use them and understand their derivation using algebra, but am stuck on understanding the logic of deriving these equations use Calculus.
I am especially not sure what Calculus rule allows us to take the integral of both sides of an equation, but there are a few other questions as well.
Please see details below.
Details of Question:
ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity
Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into:
s − s0 = v0t + ½at2
My main question is about the integration of both sides of the equation ds=v dt: First off, is this a differential equation? If not, what type of equation is this? Going back, we first multiply dt on both sides of ds/dt= v by dt to get ds= v dt and then we integrate. 1) what math rule allows us to multiply dt to both sides of this equation? 2) What calculus rule allows us to take the integral of both sides of this equation? 3) How can I visualize that these 2 integrals as being equal? 3) Would a one year college Calculus class be enough to understand this? I took 1 year of college Calculus but don't recall learning about this.
s
⌠
⌡ ds
s0
equals
t
⌠
⌡ (v0 + at) dt
0
Thanks for any help!
ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity
Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into:
s − s0 = v0t + ½at2
My main question is about the integration of both sides of the equation ds=v dt: First off, is this a differential equation? If not, what type of equation is this? Going back, we first multiply dt on both sides of ds/dt= v by dt to get ds= v dt and then we integrate. 1) what math rule allows us to multiply dt to both sides of this equation? 2) What calculus rule allows us to take the integral of both sides of this equation? 3) How can I visualize that these 2 integrals as being equal? 3) Would a one year college Calculus class be enough to understand this? I took 1 year of college Calculus but don't recall learning about this.
s
⌠
⌡ ds
s0
equals
t
⌠
⌡ (v0 + at) dt
0
Thanks for any help!