Solve for t h=-1/2at^2+1

$h = - \frac{1}{2} a t^{2} + 1$

Step 1

Rewrite the equation as $- \frac{1}{2} \cdot (a t^{2}) + 1 = h$ .

$- \frac{1}{2} \cdot (a t^{2}) + 1 = h$

Step 2

Multiply $- \frac{1}{2} (a t^{2})$ .

Combine $a$ and $\frac{1}{2}$ .

$- \frac{a}{2} t^{2} + 1 = h$

Combine $t^{2}$ and $\frac{a}{2}$ .

$- \frac{t^{2} a}{2} + 1 = h$

Step 3

Subtract $1$ from both sides of the equation.

$- \frac{t^{2} a}{2} = h - 1$

Step 4

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{t^{2} a}{2}) = - 2 (h - 1)$

Step 5

Simplify both sides of the equation.

Simplify the left side.

Simplify $- 2 (- \frac{t^{2} a}{2})$ .

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{t^{2} a}{2}$ into the numerator.

$- 2 \frac{- t^{2} a}{2} = - 2 (h - 1)$

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- t^{2} a}{2} = - 2 (h - 1)$

Cancel the common factor.

$2 \cdot - 1 \frac{- t^{2} a}{2} = - 2 (h - 1)$

Rewrite the expression.

$- (- t^{2} a) = - 2 (h - 1)$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 (t^{2} a) = - 2 (h - 1)$

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

$t^{2} a = - 2 (h - 1)$

Simplify the right side.

Simplify $- 2 (h - 1)$ .

Apply the distributive property.

$t^{2} a = - 2 h - 2 \cdot - 1$

Multiply $- 2$ by $- 1$ .

$t^{2} a = - 2 h + 2$

Step 6

Divide each term in $t^{2} a = - 2 h + 2$ by $a$ and simplify.

Divide each term in $t^{2} a = - 2 h + 2$ by $a$ .

$\frac{t^{2} a}{a} = \frac{- 2 h}{a} + \frac{2}{a}$

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{t^{2} a}{a} = \frac{- 2 h}{a} + \frac{2}{a}$

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{- 2 h}{a} + \frac{2}{a}$

Simplify the right side.

Move the negative in front of the fraction.

$t^{2} = - \frac{2 h}{a} + \frac{2}{a}$

Step 7

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{- \frac{2 h}{a} + \frac{2}{a}}$

Step 8

Simplify $\pm \sqrt{- \frac{2 h}{a} + \frac{2}{a}}$ .

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{- 2 h + 2}{a}}$

Factor $2$ out of $- 2 h + 2$ .

Factor $2$ out of $- 2 h$ .

$t = \pm \sqrt{\frac{2 (- h) + 2}{a}}$

Factor $2$ out of $2$ .

$t = \pm \sqrt{\frac{2 (- h) + 2 (1)}{a}}$

Factor $2$ out of $2 (- h) + 2 (1)$ .

$t = \pm \sqrt{\frac{2 (- h + 1)}{a}}$

Rewrite $\sqrt{\frac{2 (- h + 1)}{a}}$ as $\frac{\sqrt{2 (- h + 1)}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{2 (- h + 1)}}{\sqrt{a}}$

Multiply $\frac{\sqrt{2 (- h + 1)}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{2 (- h + 1)}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 (- h + 1)}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{\sqrt{a} \sqrt{a}}$

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{{\sqrt{a}}^{2}}$

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{a^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{a^{\frac{2}{2}}}$

Rewrite the expression.

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{a^{1}}$

Simplify.

$t = \pm \frac{\sqrt{2 (- h + 1)} \sqrt{a}}{a}$

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{2 (- h + 1) a}}{a}$

Reorder factors in $\pm \frac{\sqrt{2 (- h + 1) a}}{a}$ .

$t = \pm \frac{\sqrt{2 a (- h + 1)}}{a}$

Step 9

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{2 a (- h + 1)}}{a}$

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{2 a (- h + 1)}}{a}$

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{2 a (- h + 1)}}{a}$

$t = - \frac{\sqrt{2 a (- h + 1)}}{a}$

$t = \frac{\sqrt{2 a (- h + 1)}}{a}$

$t = - \frac{\sqrt{2 a (- h + 1)}}{a}$

Solve for t h=-1/2at^2+1

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