Question

Kilauea in Hawaii is the world’s most continuously active volcano. When we say very active volcanoes, they normally eject red-hot rocks and lava instead of smoke and ash. Suppose at an angle 35.0⁰ above the horizontal; a large rock is ejected from the volcano with a speed of 25 𝑚/𝑠. The rock hits the side of the volcano at an altitude of 20.0 𝑚 lower than its starting point. (a) Calculate the time it takes them to follow its path. (b) What are the magnitude and the direction of the rock’s velocity at impact?

Answer 1

(a) Find the time to reach the highest point:

$$t_u=\frac{v\sin35°}{g}=1.46\text{ s}.$$

The time to fall from the highest point:

$$t_d=\sqrt{\frac{2(d+v^2\sin^2\theta/(2g))}{g}}=2.49\text{ s}.$$

The total time:

$$T=t_u+t_d=3.95\text{ s}.$$

(b) The horizontal and vertical components of velocity will help us to find the resultant velocity at the impact:

$$v_x=v\cos\theta=20.5\text{ m/s}.\\ v_y=gt_d=24.4\text{ m/s}.\\ v=\sqrt{v_x^2+v_y^2}=31.9\text{ m/s}.$$