Solve for t s=-16t^2+v_0t

$s = - 16 t^{2} + v_{0} t$

Step 1

Rewrite the equation as $- 16 t^{2} + v_{0} t = s$ .

$- 16 t^{2} + v_{0} t = s$

Step 2

Subtract $s$ from both sides of the equation.

$- 16 t^{2} + v_{0} t - s = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = - 16$ , $b = v_{0}$ , and $c = - s$ into the quadratic formula and solve for $t$ .

$\frac{- v_{0} \pm \sqrt{{v_{0}}^{2} - 4 \cdot (- 16 \cdot (- s))}}{2 \cdot - 16}$

Step 5

Simplify.

Multiply $- 4 \cdot - 16 \cdot - 1$ .

Multiply $- 4$ by $- 16$ .

$t = \frac{- v_{0} \pm \sqrt{{v_{0}}^{2} + 64 \cdot (- 1 s)}}{2 \cdot - 16}$

Multiply $64$ by $- 1$ .

$t = \frac{- v_{0} \pm \sqrt{{v_{0}}^{2} - 64 s}}{2 \cdot - 16}$

Multiply $2$ by $- 16$ .

$t = \frac{- v_{0} \pm \sqrt{{v_{0}}^{2} - 64 s}}{- 32}$

Simplify $\frac{- v_{0} \pm \sqrt{{v_{0}}^{2} - 64 s}}{- 32}$ .

$t = \frac{v_{0} \pm \sqrt{{v_{0}}^{2} - 64 s}}{32}$

Step 6

The final answer is the combination of both solutions.

$t = \frac{v_{0} + \sqrt{{v_{0}}^{2} - 64 s}}{32}$

$t = \frac{v_{0} - \sqrt{{v_{0}}^{2} - 64 s}}{32}$