How long does it take for a projectile with the speed-time equation $y=x^2 + 2$ to reach speed $15$ m/s?

Question

I've been given the following equation to graph, plus a couple of sub-problems I need to solve, and I'm having trouble solving one of them. A quick summary of the problem:

A speeding projectile observed at time $0$ can be represented by the graph $y = x^2 + 2$ for the range of $-2$ seconds to $4$ seconds. a) Plot the graph (already done). b) How long does it take to reach speed $15$ m/s.

It is b) that I'm having the problem with. I thought this would be as simple as taking the equation and turning it into \begin{align*} 15 &= x^2 + 2\\ 13 & = x^2\\ \sqrt{13} & = 3.61\\ x & = 3.61 \end{align*}

But I'm not sure if I am doing this right. Am I missing something here, or is this correct? $X$ and $Y$ represent time and speed, respectively.

Answer 1

From your problem, you stated that y is the speed in m/s of the projectile and x is the time elapsed. Given those conditions, your solution would be correct. It merely asks at what x value (time) is the speed (y value) equal to 15 m/s, so you just solve the equation for for 15 m/s and that's it. I assume that the starting time is x = 0 because you said is observed a time 0.

Answer 2

I would guess that $y$ represents the position and $x$ the time. If this is the case, then the speed would be given by $\frac{d y}{dx} = 2 x$. So, the required speed is reached when $\frac{d y}{dx} = 15$, or $x=7.5$. How long it takes depends on the initial time, so the answer is $7.5-x_0$, where $x_0$ is the initial time.

However, this is a guess, since I do not know what $x$ and $y$ represent.