Question

Volcanic Lava Fountains Although the November 1959 Kilauea Iki eruption on the island of Hawaii began with a line of fountains along the wall of the crater, activity was later confined to a single vent in the crater's floor, which at one point shot lava 1900 ft straight into the air (a world record). What was the lava's exit velocity in feet per second? in miles per hour? [Hint: If \(v_{0}\) is the exit velocity of a particle of lava, its height \(t\) seconds later will be \(s=v_{0} t-16 t^{2}\) feet. Begin by finding the time at which \(d s / d t=0 .\) Neglect air resistance.

Short answer

The lava's exit velocity is approximately 348.9 feet per second or 237.6 miles per hour.

Step-by-step solution

Calculate Time

Find the time at which ds/dt = 0. This is when the lava reaches it's maximum height. To do this, differentiate the equation of height respect to time, \(s=v_{0} t-16 t^{2}\), to get the velocity as a function of time. Take the derivative to obtain ds/dt = \( v_{0} - 32t \). Set this to zero and solve for t, yielding \( t= \frac{v_{0}}{32} \).

Solve for Initial Velocity

Now that we know the time, t, we can substitute it back into the original equation to find the initial velocity. Substituting \(t = \frac{v_{0}}{32}\) the equation becomes \(s = \frac{v_0^2}{32} - 16*\frac{v_0^2}{32^2}\). Simplify this equation to \(s = \frac{v_0^2}{64}\) and solve for the initial velocity, \(v_{0}\) by multiplying both sides by 64 to get \(64s = v_0^2\), then take the square root of both sides to isolate \(v_0\), resulting in \( v_{0} = 8 \sqrt{s}\).

Substituting Specific Value

The problem states that the lava shot 1900 ft straight into the air. Substitute this value of s into the formula found in the previous step, \( v_{0} = 8 \sqrt{s}\), to find the initial velocity in feet per second. This gives \(v_{0} = 8 \sqrt{1900}\) feet per second.

Unit Conversion

The problem also asks for the exit velocity in miles per hour. To convert feet per second to miles per hour, multiply by the conversion factor 3600/5280. Calculating \(v_{0} = 8 \sqrt{1900} * \frac{3600}{5280}\) gives \(v_{0}\) in miles per hour.