Solve for t h(t)=-16t^2+v_0t+h_0

$h (t) = - 16 t^{2} + v_{0} t + h_{0}$

Step 1

Since $t$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Subtract $h t$ from both sides of the equation.

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

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

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

Step 5

Simplify.

Simplify the numerator.

Apply the distributive property.

$t = \frac{- v_{0} + h \pm \sqrt{{(v_{0} - h)}^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $- - h$ .

Multiply $- 1$ by $- 1$ .

$t = \frac{- v_{0} + 1 h \pm \sqrt{{(v_{0} - h)}^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $h$ by $1$ .

$t = \frac{- v_{0} + h \pm \sqrt{{(v_{0} - h)}^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Rewrite ${(v_{0} - h)}^{2}$ as $(v_{0} - h) (v_{0} - h)$ .

$t = \frac{- v_{0} + h \pm \sqrt{(v_{0} - h) (v_{0} - h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Expand $(v_{0} - h) (v_{0} - h)$ using the FOIL Method.

Apply the distributive property.

$t = \frac{- v_{0} + h \pm \sqrt{v_{0} (v_{0} - h) - h (v_{0} - h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- v_{0} + h \pm \sqrt{v_{0} (v_{0}) + v_{0} (- h) - h (v_{0} - h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Apply the distributive property.

$t = \frac{- v_{0} + h \pm \sqrt{v_{0} (v_{0}) + v_{0} (- h) - h v_{0} - h (- h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Simplify and combine like terms.

Simplify each term.

Multiply $v_{0}$ by $v_{0}$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} + v_{0} (- h) - h v_{0} - h (- h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Rewrite using the commutative property of multiplication.

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} - h (- h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Rewrite using the commutative property of multiplication.

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} - 1 \cdot (- 1 h \cdot h) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $h$ by $h$ by adding the exponents.

Move $h$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} - 1 \cdot (- 1 (h \cdot h)) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $h$ by $h$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} - 1 \cdot (- 1 h^{2}) - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $- 1$ by $- 1$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} + 1 h^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $h^{2}$ by $1$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - v_{0} h - h v_{0} + h^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Subtract $h v_{0}$ from $- v_{0} h$ .

Move $v_{0}$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - h v_{0} - h v_{0} + h^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Subtract $h v_{0}$ from $- h v_{0}$ .

$t = \frac{- v_{0} + h \pm \sqrt{{v_{0}}^{2} - 2 h v_{0} + h^{2} - 4 \cdot - 16 \cdot h_{0}}}{2 \cdot - 16}$

Multiply $- 4$ by $- 16$ .

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

Multiply $2$ by $- 16$ .

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

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

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

Step 6

The final answer is the combination of both solutions.

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

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

Solve for t h(t)=-16t^2+v_0t+h_0

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