Solving for Displacement given Velocity and Acceleration?

Question

I have given a problem to solve for the displacement of an object given the acceleration $-10\;\mathrm{m/s^2}$ and the initial velocity $627\;\mathrm{m/s}$ of how far an object shoots up, though I am not given the time, so I can't use the equation

$$S=vt+\frac{at^2}2$$

Is there a way to solve this? Thanks!

Edit Thanks to whoever edited my question for the formatting

Answer 1

You can use Torricelli's equation: $$v_f^2 = v_0^2+2a\Delta S$$ where $v_f, v_0$ are, respectively, the final and initial velocity of the object, $a$ its acceleration and $\Delta S$ its displacement, and solve for $\Delta S$.

You know that at the point of maximum height, the object's velocity is 0 $m/s$. Substituting the other terms, and solving for $\Delta S$, we have $$\Delta S = \frac{-v_0^2}{2a} = \frac{-627^2}{-20} = 19 656.45 m$$

Answer 2

If acceleration is $-10$, and velocity is an antiderivative of acceleration, then $$ v(t) = \int -10 \, dt = -10t + C. $$ If you put in $t = 0$ you will see that $C$ is your initial velocity. Then you have that displacement is an antiderivative of velocity.