# Why should I find value of constant of integration instead of using definite integration?

$$v = 2t$$, find $$s$$ (displacement) at 3 seconds if at $$t=0$$ body is 2m behind origin.

My teacher said that I cannot use definite integration as we do not know the boundaries for time, so I have to first use indefinite integration and find the value of c. Apparently, I have to do the following steps:

$$s = \int v \space dt$$ $$s = \int (2t) \space dt$$ $$s= {t^2} + c$$

Now, since we are given that initially (at $$t=0$$), the body is 2m behind origin (the displacement is apparently -2m), we plug in $$t=0$$ and $$s=-2$$ into the above equation, getting $$c=-2$$. Next, I have to plug in $$c=-2$$ and $$t=3$$ into the above equation, to find the required answer.

1. Just because the body is starting from 2m behind the origin, does that mean the displacement at $$t=0$$ is -2m? What does that even mean? I'm confused about the difference between position and displacement
2. Shouldn't the boundaries be $$t=0$$ and $$t=3$$ ? But when I use those as boundaries and try to solve using the definite integral, vs. when I use the indefinite integral method shown above, my answer differs by $$c$$. What does this mean?
3. What does it mean to "find displacement at 3 seconds"?
4. What exactly is the difference between asking to "find displacement at 3 seconds" and "find displacement from 0 to 3 seconds"? In the latter I can use definite integration by putting $$t=0$$ and $$t=3$$ as the boundaries, and can't do so in the former, so there must be some difference.
• you can find $\Delta s = \int_0^3 2t dt=(t^2)|_0^3=9$. So add the change in displacement to the initial position and you get 7. Jul 30, 2022 at 19:00

3. Doesn't quite make sense in our normal language. You should say the displacement between 2 time intervals. I believe by the displacement at ($$t=0$$) you mean the final position. So similar to 2., you add the displacement to the initial position to find the final position.
• Thank you so much, this is a genuinely perfect answer to me! So in conclusion, adding $c$ to my final answer after evaluating the definite integral is done because the integral itself will only give me displacement, which is change in position, but I want to find the final position. Correct?