Question

During volcanic eruptions, chunks of solid rock can be blasted out of the volcano; these projectiles are called volcanic bombs.Figure 4-51 shows a cross section of Mt. Fuji, in Japan. (a) At what initial speed would a bomb have to be ejected, at angle${\mathbf{\theta }}_{\mathbf{0}}\mathbf{=}\mathbf{35}\mathbf{°}$to the horizontal, from the vent at Ain order to fall at the foot of the volcano at B,at vertical distance h=3.30km and horizontal distance$\mathrm{d}=9.40\mathrm{km}$? Ignore, for the moment, the effects of air on the bomb’s travel. (b) What would be the time of flight? (c) Would the effect of the air increase or decrease your answer in (a)?

Short answer

a) The initial velocity of a volcanic bomb is 260 m/s

b) The time of flight is 45 s.

c) The initial velocity will be increase.

Step-by-step solution

The given data

$$\begin{gathered} 1)\mathrm{Angle}\mathrm{of}\mathrm{the}\mathrm{ejection},\mathrm{\theta }=35° \\ 2)\mathrm{Vertical}\mathrm{distance},\mathrm{y}=3.30\mathrm{km}\mathrm{or}-3300\mathrm{m}\left(\mathrm{downard}\right) \\ 3)\mathrm{Horizontal}\mathrm{distance},\mathrm{x}=9.40\mathrm{km}\mathrm{or}9400\mathrm{m} \end{gathered}$$

Understanding the concept of the relative motion

The volcanic bomb has only gravitational force acting on it. Therefore, its acceleration is constant in the vertical direction and there is no acceleration in the horizontal direction. Hence, we can treat this motion as projectile motion.

Formulae:

The equation of distance in the projectile motion,$\mathrm{y}={\mathrm{tan\theta }}_0\mathrm{x}-\frac{{\mathrm{gx}}^2}{2{\left({\mathrm{V}}_0{\mathrm{cos\theta }}_0\right)}^2}$ …(i)

The time of flight of a body in projectile motion $\mathrm{T}=\frac{\mathrm{x}}{{\mathrm{V}}_0{\mathrm{cos\theta }}_0}$ …(ii)

(a) Calculation of the initial velocity

By rearranging equation (i) of projectile motion, we get the value of the initial velocity of the volcanic bomb by substituting the given values as follows:

$$\begin{gathered} {\mathrm{V}}_0=\frac{9400\mathrm{m}}{\cos \left(35°\right)}\sqrt{\frac{9.8\mathrm{m}/{\mathrm{s}}^2}{2\left(9400\mathrm{m}\times \tan (35°\right)-\left(-3300\mathrm{m})\right)}} \\ =255.52\mathrm{m}/\mathrm{s} \\ \approx 260\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of the initial velocity is 260 m/s.

(b) Calculation of the time of flight

Using the given values in equation (ii), we can get the value of the time of flight as follows:

$$\begin{gathered} \mathrm{T}=\frac{9400\mathrm{m}}{\left(255.52\mathrm{m}/\mathrm{s}\right)\cos \left(35°\right)} \\ =44.91\sec \\ \approx 45\sec \end{gathered}$$

Hence, the value of time is 45 se c.

(c) Calculation of the effect of the air

The air will oppose the motion of the volcanic bomb, hence we expect the initial speed required to reach the foot of the volcano will be increase.