Question

an engineer is planning a firework show. he knows that the firework must reach a minimum height of 240m for safety reasons. neglect drag. what is the launch speed required for the firework to reach critical height (in given units)? mph could the firework be safely launched with a speed of 140mph?

Explanation:

Step1: Use kinematic - energy conservation

The initial kinetic energy $K_{i}=\frac{1}{2}mv_{0}^{2}$ and at the maximum - height, the final kinetic energy $K_{f} = 0$ and the potential energy $U_{f}=mgh$. By conservation of mechanical energy $K_{i}=U_{f}$, so $\frac{1}{2}mv_{0}^{2}=mgh$.

Step2: Solve for the initial velocity $v_{0}$

Canceling out the mass $m$ from both sides of the equation $\frac{1}{2}v_{0}^{2}=gh$. We know that $g = 9.8m/s^{2}$ and $h = 240m$. Then $v_{0}=\sqrt{2gh}$.
$v_{0}=\sqrt{2\times9.8\times240}=\sqrt{4704}\approx68.6m/s$.

Step3: Convert $m/s$ to $mph$

We know that $1m/s = 2.23694mph$. So $v_{0}=68.6\times2.23694\approx153.5mph$.

Answer:

153.5 mph