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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)? Could the firework be safely launched with a speed of 160mph?Hint: Work the problem in metric units (m, m/s and m/s2) and then convert the initial speed from m/s to mph. Speeds that are greater than the critical speed you calculated are safe, because they cause the firework to safely explode higher than the critical height. We do NOT want the firework to explode too close to the ground!

An Engineer Is Planning A Firework Show He Knows That The Firework Must Reach A Minimum Height Of 240m For Safety Reasons Neglect DragWhat Is The Launch Speed R class=

Sagot :

We are given that a firework is launched from the ground and it is required that it reaches a height of 240m. Since we are ignoring the drag we can use the equation of motion of an object in free fall. Specifically, we will use the formula for the maximum height of an object:

[tex]H=\frac{v^2_0}{2g}[/tex]

Where:

[tex]\begin{gathered} H\text{ = height} \\ v_0=\text{ initial sp}ed \\ g\text{ = acceleration of gravity} \end{gathered}[/tex]

We solve for the initial speed first by multiplying both sides by "2g":

[tex]2gH=v^2_0[/tex]

Now we take the square root to both sides:

[tex]\sqrt[]{2gH}=v_0[/tex]

Now we replace the given values:

[tex]\sqrt[]{2(9.8\frac{m}{s^2})(240m)}=v_0[/tex]

Solving the operations:

[tex]68.6\frac{m}{s}=v_0[/tex]

Since we are asked to determine the speed in miles per hour we will use the following conversion factors:

[tex]\begin{gathered} 1\text{ mile = }1609.34\text{ m}eters \\ 1\text{ hour = 3600 seconds} \end{gathered}[/tex]

Now we multiply the speed by the conversion factors in the form of fractions:

[tex]68.6\frac{m}{s}\times\frac{1\text{mile}}{1609.34m}\times\frac{3600s}{1h}[/tex]

Solving the operation:

[tex]v_0=153.42\text{ mph}[/tex]

Therefore, the launch speed must be 153.42 mph.

If the velocity is 160 miles per hour it would be safe because a larger speed would increase the maximum height Since the height is proportional to the square of the speed.

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