Question

1.
A) Mountain rescue officers trigger an avalanche by firing a gun at a distant mountain slope. The muzzle velocity of the gun is 190 m/s. The gun muzzle is at sea level (y=0). The horizontal and vertical distance from the gun to the slope is 640 m and 240 m, respectively. Find the lowest angle between the barrel of the gun and the horizontal plane.
Round your answer to the nearest tenth of degree.

B) The mountain rescue officers trigger the avalanche by firing a gun at a distant mountain slope. The muzzle velocity of the gun is 140 m/s. The gun is at the sea-level (y=0). The horizontal and vertical distance from the gun to the slope is 790 m and 210 m, respectively. Find the highest angle between the barrel of the gun and the horizontal plane that results in direct hit.
Round your answer to the nearest tenth of degree.

Answer 1

PROJECTILE

along horizontal
________________

initial velocity vox = vo*costheta

acceleration ax = 0

initial position = xo = 0

final position = x

displacement = x - x0

from equation of motion

x - x0 = v0x*T+ 0.5*ax*T^2

x - x0 = vo*costheta*T

T = (x - x0)/(vo*costheta)......(1)

along vertical
______________

initial velocity v0y = vo*sintheta

acceleration ay = -g = -9.8 m/s^2

initial position y0 = 0

final position y = 0

from equation of motion

y-y0 = v0y*T + 0.5*ay*T^2 .........(2)

using 1 in 2

y-y0 = (vo*sin(theta)*(x-x0))/(vo*cos(theta)) - (0.5*g*(x-x0)^2)/(vo^2*(cos(theta))^2)

y - y0 = (x - x0)*tantheta - (0.5*g*(x-x0)^2)/(vo^2*(cos(theta))^2)

costheta = a

sintheta = sqrt(1-a^2)

tantheta = sqrt(1-a^2)/a

y - y0 = (x - x0)*sqrt(1-a^2)/a - (0.5*g*(x-x0)^2)/(vo^2*a^2)

A)

y = 240 m

x = 640 m

vo = 190 m/s

240 - 0 = (640 - 0)*sqrt(1-a^2)/a - (0.5*9.8*640^2/(190^2*a^2))

a = 0.0903 or 0.9007

costheta = 0.0903 or 0.9

for minimum angle

costheta = 0.9

theta = cos^-1(0.9) = 25.84 degrees

==============================

B)

y = 210 m

x = 790 m

vo = 140 m/s

210 - 0 = (790 - 0)*sqrt(1-a^2)/a - (0.5*9.8*640^2/(140^2*a^2))

a = 0.136 or 0.923

costheta = 0.0903 or 0.9

for highest angle

costheta = 0.136

theta = cos^-1(0.136) = 82.18 degrees