Solving a Volcanic Eruption Problem with Constant Acceleration Equations

[martyk]

Greetings,
Here is my following question

1. During volcanic eruptions, chunks of solid rock can be blasted out of the volcano; these projectiles are called volcanic bombs. Figure 4-58 shows a cross section of Mt. Fuji, in Japan. (a) At what initial speed would a bomb have to be ejected, at angle θ0 = 35˚ to the horizontal, from the vent at A in order to fall at the foot of the volcano at B, at vertical distance h = 2.90 km and horizontal distance d = 9.00 km? Ignore, for the moment, the effects of air on the bomb's travel. (b) What would be the time of flight?



2. y = (tan theta)x - (gx^2) / 2(V0costheta0)^2



3. The above equation listed is the only thing that I can think of that works, usually constant acceleration equations aren't seem to getting anywhere for me. But anyways, I solved the equation to just the initial velocity (V0), just wanted you guys' opinions on whether this is the right approach to the problem

Thanks,

 

[jambaugh]

I would start with the two components of motion, x(t) and y(t). Recall the equations of motion are the position equals the initial position plus initial velocity times time plus half the acceleration times the square of the time. Since the only acceleration is gravity only the y component has that term.

Break the initial velocity into components using trigonometry. Since the angle given is off of the horizontal the horizontal component is proportional to the cosine and the vertical proportional to the sine. (In general just remember cosine is 1 when angle is zero so cosine is associated with the direction off of which the angle is measured.)

$$x(t) = x_0 + V_0 cos(35^o)\cdot t$$
$$y(t) = y_0 + V_0 sin(35^o) \cdot t - (g/2)t^2$$
where g is 9.8 m/s^2 is the gravitational acceleration.

You have two equations and two unknowns (V0 and t) since you know the initial and final positions. Solve and be sure to keep track of your units.

 

[tomcue]

Hello,

Thank you for sharing your question and solution to the volcanic eruption problem using constant acceleration equations. I would like to provide my perspective on your approach to this problem.

Firstly, using constant acceleration equations can be a valid approach to solving this problem, as long as certain assumptions are made and limitations are taken into consideration. For example, as you mentioned, the effects of air resistance on the bomb's travel are ignored in this problem. This may not accurately reflect the real-world scenario, as air resistance can significantly affect the trajectory of the bomb. Therefore, it is important to acknowledge and discuss the limitations of using constant acceleration equations in this case.

Secondly, it is important to carefully choose and apply the correct equation for the given problem. In this case, the equation you provided (y = (tan theta)x - (gx^2) / 2(V0costheta0)^2) seems to be the correct one to use, as it takes into account the initial angle and velocity of the bomb, as well as the acceleration due to gravity.

However, it is also important to consider other factors that may affect the bomb's trajectory, such as the shape and size of the bomb, wind conditions, and the terrain of the volcano. These factors may not be easily accounted for in the equation you provided, and may require further analysis and calculations.

In conclusion, while using constant acceleration equations can be a valid approach to solving this problem, it is important to acknowledge the limitations and carefully consider all relevant factors to accurately model the trajectory of the volcanic bomb. I hope this helps and good luck with your problem-solving!