Motion of projectile?

Question

Volcanoes on the Earth eject rocks at speeds of up to 110 m/s. Consider a 1000 m high volcano which ejects rocks in all directions. What is the maximum height above sea level reached by rocks?




What is the maximum horizontal distance at sea level reached by the rocks?

Answer 1

a) V=gt and H=0.5 g t^2

h= 0.5 g (V/g)^2

H= 1000+ h=
H= 1000 + (V/)^2 /(2 g)
H= 1620 m

The maximum horizontal distance at sea level reached by the rocks depends on the horizontal component and it was not specified. Coriolis effect is negligible.

A fanout can be estimated if the geometry of the vent is known and the size of the caldera. Typical fanout can be assumed at 10 to 15 degrees. Then the we have to resolve the initial velocity into vertical and horizontal components.
Vv= V cos(15) using max angle
Vh= Vsin(15)

Then
H= 1000 + (Vcos(15))^2 /(2 g)
H= 1000 + 575= 1575m
S=Vsin(15) t where
t= t1 + t2
t1- time to travel to the upper most point h=(Vcos(15))^2 /(2 g)

t2 - time to fall height H

In general t=sqrt(2h/g)
t1= sqrt(2h/g)= sqrt(2 x 575 / 9.81)=11 s
t2= sqrt(2H/g) = sqrt(2 x 1575 /9.81)=18s
t= 29 s

S= V sin(15) t=
S= 110 sin(15) x 29= 830m max for a 15 degree fanout.

Answer 2

To answer the first, use the kinematics equation

2 g h = vo²

g = 9.8 m/s², vo = 110 m/s, and h is what you're looking for.

For the second, know that in the simple approximations typical for these problems (no air resistance, ignore the rotation of the Earth, and so on) you get the maximum range at an angle of 45 degrees. So this time use the first equation but with vo = ( 110 m/s ) ( sin 45 degrees ) and solve for height. Then use

h = vo t - 1/2 g t²

to find elapsed time. Finally,

range = horizontal velocity x time

where the horizontal velocity is ( 110 m/s ) ( cos 45 degrees )

Answer 3

maximum height would be straight up (vertical)..

v(final) = v(original) + a*t
0 = 110 + (-9.8)*t
t = 11.22 sec. to reach peak

y = v(original)*t + (1/2)*a*t^2 + 1000
y = (110)(11.22) + (.5)(-9.8)(11.22)^2 + 1000
y = 1234.69379 - 617.34685 + 1000
y = 1617.35 meters


for furthest horizontal distance, rock shoots at 45 degree angle...
y = (v)(t) + (1/2)(a)(t^2)
-1000 = (110*sin 45)(t) + (1/2)(-9.8)(t^2)... you can do the rest (i dont have a graphing calculator on me)