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1

Solved example of physics

With what speed should a stone be thrown upward so that it reaches a maximum height of 3.2 m?
2

What do we already know? We know the values for acceleration ($a$), velocity ($v$), distance ($y$), height ($y_0$) and want to calculate the value of velocity ($v_0$)

$a=9.81\:m/s2,\:\: v=0,\:\: y=3.2\:m,\:\: y_0=0,\:\: v_0=\:?$
3

According to the initial data we have about the problem, the following formula would be the most useful to find the unknown ($v_0$) that we are looking for. We need to solve the equation below for $v_0$

$v^2=v_0^2-2a\left(y- y_0\right)$

4

We substitute the data of the problem in the formula and proceed to simplify the equation

$0^2=v_0^2-2\cdot 9.81\cdot \left(3.2- 0\right)$
5

Multiply $-2$ times $9.81$

$0^2=v_0^2-19.62\cdot 3.2$
6

Multiply $-19.62$ times $3.2$

$0^2=v_0^2-62.784$
7

Calculate the power $0^2$

$0=v_0^2-62.784$
8

Rearrange the equation

$v_0^2-62.784=0$

We need to isolate the dependent variable $v_0$, we can do that by simultaneously subtracting $-62.784$ from both sides of the equation

$v_0^2=0+62.784$

$x+0=x$, where $x$ is any expression

$v_0^2=62.784$
9

We need to isolate the dependent variable $v_0$, we can do that by simultaneously subtracting $-62.784$ from both sides of the equation

$v_0^2=62.784$

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$

$\left(v_0^2\right)^{\frac{1}{2}}=62.784^{\frac{1}{2}}$

Divide $1$ by $2$

$\sqrt{v_0^2}=62.784^{\frac{1}{2}}$

Simplify $\sqrt{v_0^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $0.5$

$v_0^{2\frac{1}{2}}$

Multiply $2$ times $0.5$

$v_0$

Multiply $2$ times $0.5$

$v_0=62.784^{\frac{1}{2}}$

Divide $1$ by $2$

$v_0=\sqrt{62.784}$

Calculate the power $\sqrt{62.784}$

$v_0=7.9236$
10

Removing the variable's exponent raising both sides of the equation to the power of $\frac{1}{2}$

$v_0=7.9236$
11

The complete answer is

The speed of the stone is $7.9236$ m/s

Final Answer

The speed of the stone is $7.9236$ m/s

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