How do a(t), v(t), and x(t) relate in physics equations?

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The discussion focuses on the relationships between acceleration a(t), velocity v(t), and position x(t) in physics equations. It clarifies that v(t) is derived from the integral of acceleration, leading to the equation v(t) = (α)t^2/2 + w. The position x(t) is then obtained by integrating v(t), resulting in x(t) = (α)t^3/6 + wt + k. A constant k is determined by substituting specific values, leading to the final expression for x(t) that includes adjustments based on initial conditions. The conversation highlights the importance of correctly applying calculus to derive these relationships.
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a(t)=dv/dt
v(t)=dx/dt



I thought it was just
v(t)= (alpha)t^2/2 + w
x(t)=((alpha)t^2)/6 + wt + a

But the answer is ((alpha)t^2)/6 + wt + a - 2w - 4(alpha)/3

Can someone please explain this to me...
 
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Starting from your v = .5*@*t^2/2 + W
x is the integral dt of this: x = .5*@/3*t^3 + wt + k
To evaluate the constant k, put in time 2 and x = A:
This gives you k = A - 4/3@ - 2w
Put that back in the x formula and you have the answer you are looking for, though there seems to be a t^2 instead of a t^3 in the answer.
 
Thank you very much!
 
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