Calculus problems involving motion

Question

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 $t=5$.

Answer 1

HINT: Your first problem requires you to find both displacement and velocity. So you have to solve the first order as well as the second order differential equation. For the second order differential equation, you will have 2 constants, say $A$ and $B$. Use the 2 conditions for displacement and solve for the constants.Then differentiate to get the expression for velocity.

Answer 2

Let me give you a hint for your second problem:

Again, the problem is to find the constant C after integrating from the acceleration to the velocity equation. To achieve this, you have to integrate the velocity equation again, to get the displacement equation. Now, you can use your knowledge for the displacement in the fourth second (calculate the difference of the displacement equation for t=4 and t=3) to get the c for the velocity equation. Then, simply solve it for t=5.

Answer 3

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 $B$.