For my first problem session in classical mechanics I got stuck on this problem about locomotive velocity. Here is the description:
A locomotive leaves a train station with the magnitude of its displacement along a straight track given by $s=(1.4 m/s^2)t^2$. Exactly $10$ s later, another locomotive on a parallel track passes the station with constant velocity. What minimum velocity must the second locomotive have so that it just catches the first locomotive without passing it? (Hint: Plot displacement versus time for both locomotives assuming the station is at $s=0$.)
I then proceeded to plot the graph of the first locomotive, and then I pretty much got stuck. I know that since the second locomotive has a constant velocity, its equation would be something like: $$s_2=vt ,$$ since there is no acceleration. I am supposed to choose an arbitrary time at which these two should meet and then set these two equations equal? ($s$ and $s_2$) this would give me a value for $v$, but it seems completely arbitrary.
I would appreciate some help. Thanks in advance.
Write the two displacements. $s_1=1.4t^2$ and $s_2=v(t-10)$ and set them equal to each other. Since you have a quadratic equation in $t$, you know the conditions you must have to have a solution. In other words make sure the discriminant is positive, that should give you the answer.