Question

A ball shot straight up with kinetic energy $K_0$ reaches height $h$. What height will it reach if the initial kinetic energy is doubled?

Short answer

It the initial kinetic energy is doubled then the height is also doubled

Step-by-step solution

Work done

Work done by a net force is equals to the change within the mechanical energy.
$W_{\text{net}}=\mathrm{\Delta }K$
Here $W_{\text{net}}$ is that the work done by the web force and $∆K$is change inmechanical energy.

Mechanical Energy

At the very best point, the full mechanical energy get converted into gravitationalP.E.,
$\begin{array}{r}U=K_0\end{array}$

$\begin{array}{r}mg=K_0\\ \end{array}$

$h=\frac{K_0}{mg}$
If themechanical energy is doubled, the corresponding height is calculated as follows,
$\begin{array}{r}{U}_2=2K_0\end{array}$

$\begin{array}{r}mg{h}_2=2K_0\\ \end{array}$

$h_2=\frac{2K_0}{mg}$

$\begin{array}{r}{h}_2=\frac{2K_0}{mg}\\ \end{array}$

$=2h$

Therefore, if mechanical energy is doubled the peak are going to be doubled.