[Math] Calculus problems involving motion

calculusintegrationkinematics

I've been working on the following problems, and I know how to integrate functions,but I do not know how to find the value of "c" in the examples below when finding the antiderivative. Any idea what to do? Cheers

  1. A particle travels in a straight line such that its acceleration at time t seconds is equal to $6t+1$ $m/s^2$. When $t=2$, the displacement is equals to $ 12m$ and when $t=3 $ the displacement is equals to $ 34m$.
    Find the displacement and velocity when $t=4$.

  2. A particle travels in a straight line with its acceleration at time t equal to $3t+2$ $ m/s^2$. The particle has an initial positive velocity and travels $30m$ in the fourth second.
    Find the velocity of the body when $t=5$.

Best Answer

For the first problem: you have the acccelaration is $$a(t)=6t+1 $$ Let $x(t)$ and $v(t)$ denote the displacement and the velocity respectively. Then $x''(t)=a(t)$. So integration $a(t) $ twice we end up with $$ x(t)= t^3 +\frac{t^2}{2}+At+B$$ but we have $x(2)=12$ and $x(3)=34$ Then $$ 8+ 2 +2A+B= 12$$ and $$ 27+\frac{9}{2}+3A+B=34 $$ subtracting the first equation from the second we get $ 17+\frac{9}{2} +A= 22 $, hence $A= \frac{1}{2}$.Then substitute to find $B$.

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