Constants for anti-derivatives

Question

Hey StackExchange I'm diving into integral calculus for the first time and I have a few questions about this problem.

A steel ball bearing at rest is accelerated in a magnetic field in a line with acceleration a(t) = 120t meters per second.

(a) Find the velocity and position at time t.

(b) Find its position in meters and velocity in meters per second after 0.5 second.

So I get the velocity by finding the anti-derivative of acceleration and I find position by getting the anti-derivative of velocity.

$$v(t) = 60t^2 + c$$

$$p(t) = ct + 20t^3 + c_{1}$$

I'm not sure how I would go about answering the questions though. What do I use for a constant when plugging in values? Thanks!

Answer 1

$\textbf{"at rest"}$ i.e. $v(t=0)=0$ is the important piece of information.

Also the integrating to get the position is a tad wrong with the constants. $$ p(t) = 20t^3 + ct + \textbf{c}_1 $$

and since the postion is relative measure we can treat the starting point as zero displacement and therefore $p(t=0)=0$ also.

Answer 2

First things first: when you integrate twice like you did, you will have two (generally different) constants of integration. Thus, your equations should look like

$$v(t) = 60t^2 + c_1$$ $$p(t) = 20t^3 + c_1t + c_2.$$

(An easy way to see why we should have two different constants of integration is the consider units. The values $ct$ and $c$ will have different units, so we can't add them together!)

Since the ball bearing begins at rest, $v(0) = 0$. If we agree that the ball bearing begins at position $0$ (or if we just decide that $p$ should represent the displacement of the ball bearing rather than it's absolute position), then we also have $p(0) = 0$. With these two equations, you can form a system of equations to solve for $c_1$ and $c_2$.